I fear, however, that the Court got it wrong in finding bias in this case. It has been quite a while since the Supreme Court of Canada has addressed the rule against bias and this case may provide it with an occasion to do so.
In fact, the leading cases in this area date from the early 1990s: Old St Boniface Residents Association v. Winnipeg (City),  3 S.C.R. 1170; Save Richmond Farmland Society v. Richmond (Township),  3 S.C.R. 1213; and Newfoundland Telephone Company v. Newfoundland (Public Utilities Board of Commissioners),  1 S.C.R. 623.
These cases establish that, when it comes to elected decision-makers, the rule against bias applies differently. Given that elected representatives are expected to have expressed strong views about matters of public importance, a relaxed standard must generally be applied. Accordingly, in order to make out a claim of bias against an elected representative, the applicant must demonstrate a "closed mind" on the part of the councillor. There must be an "expression of a final opinion on the matter, which cannot be dislodged" (Old St Boniface at p. 1197). However, in a situation where the elected representative is sitting in an adjudicative capacity, as a member of a decision-making committee, the usual test applies: would a reasonable observer fear impartiality (Newfoundland Telephone).
In summary, once an adjudicative hearing commences the elected representative must be more careful about what he or she says, but up to that point, the more relaxed "closed mind" standard applies.
The problem in Beaverford was that a councillor who was opposed in general to quarrying in a geographic area sat as a member of a municipal committee which refused a gravel extraction permit to the applicant company:
 To support his bias allegation, counsel for Beaverford provided the SDAB panel with copies of postings from Croswell’s Facebook page. These postings contained pictures of a different gravel pit, near where Beaverford’s proposed pit would be located. Croswell openly opposed that development by 541466 Alberta Ltd., the same company that Beaverford had retained as a consultant. Croswell had added a photo caption that described the site in part as “a waste land for private profit”. Counsel for Beaverford also produced a flier and an open letter that Croswell had written to his constituents. The flier explained that Croswell had introduced a motion at his last council meeting suggesting that the County prohibit any further gravel extraction, unless it was for the County’s own use. The letter expressed Croswell’s concern over the approval process utilized for the specific gravel pit that he had posted on Facebook, noting that “the clear‑cutting of a pristine, forested area near the lake has already commenced”. Croswell’s comments also questioned why the County of Thorhild was approving gravel pits and, effectively, giving away millions of dollars worth of gravel when the County may require use of that resource in the future. Croswell’s Facebook postings, the flier and the letter were issued in March, 2010. Beaverford’s appeal before the SDAB was heard on September 22, 2011...
The Court's reasoning is difficult to follow. Read generously, paras. 25 and 26 indicate that the Court had the correct test in mind: closed mind standard up to the commencement of the hearing; reasonable apprehension of bias afterward. [O]n two occasions in July, 2011, notably on July 26, 2011, and thus much closer to the date of the SDAB hearing than the motion evidence, Croswell moved that the Land Use Bylaw 989‑98 be amended to prohibit development for aggregate extraction purposes on any Crown land within the County of Thorhild No. 7 other than for the use of the County or Provincial and Federal Government Transportation requirements. He went on to suggest a fettering of future decisions by suggesting that exception to this prohibition would require unanimous council approval.
However, the Court stated the closed mind test in the following terms:
This is confusing. In Old St Boniface, the Supreme Court of Canada stated that it is for the court to form an opinion on whether the decision-maker had an open mind (at p. 1197). There is no mention of whether a reasonable person would think the decision-maker had an open mind. The standard applied by the Court is thus more stringent. Would a reasonable person, knowledgeable of the facts, and having thought the matter through, conclude that Croswell had a settled opinion against developments such as the applicant’s prior to SDAB hearing? Since there is both an attitudinal and behavioural aspect to lack of impartiality, the Court would as part of the analysis consider whether a reasonable person could have confidence that Croswell would approach the matter with an open mind.  In our view, while the open mind test may apply to tribunals with a policy or legislative assignment, the traditional reasonable apprehension of bias test should apply to the SDAB when executing its quasi-judicial function...
Also confusing is the Court's reference to the doctrine of necessity in its assessment of whether a more relaxed standard for bias was appropriate:
However, the doctrine of necessity is an exception to the rule against bias, applied only after a court has concluded that a decision-maker was biased. If there were no other means of taking a decision, in such a way as to avoid the participation of a decision-maker tainted by bias, then the decision has to stand. But here the Court used considerations of necessity to justify its imposition of a stringent standard for bias. With respect, this reasoning puts the necessity cart before the bias horse. The fact that Croswell did not need to participate is simply a red herring. Nevertheless, another of the practicalities of local governance is that it is not always necessary for a person who has acted as a strong advocate for a position directly related to the subject matter before the SDAB to participate in the matter, where other equally qualified participants in the SDAB hearing are available. It was not disputed before this Court that the local SDAB had not run out of qualified participants. That factual reality is significant here. It means that there was no necessity for Croswell to have taken part in this SDAB hearing, that necessity concept being reflected in Peters v Strathcona (County No 20), 1989 ABCA 313 (CanLII), (1989) 102 AR 241 (CA) at paras 6 to 8.
Ultimately, the Court held that Croswell's previous legislative activities demonstrated that he had a closed mind on the subject at hand:
 The lack of confidence of an open mind in the face of a clearly adverse attitude reflected in a history of adverse behaviour is, in our view, sufficient in this case to find that a reasonable and informed person who thought the matter through would have a reasonable apprehension of bias in Croswell. As said, saying so does not mean the open mind standard should be the test. It merely means that proof of an inability to approach the matter with an open mind here is sufficient...
 Specifically, the wording of the July, 2011, resolutions proposed by Croswell were not only to “prohibit” gravel extractions in the County, but even to bind future councils by a requirement that only a unanimous decision could change the prohibition later. It was only a matter of weeks between these resolutions and Croswell’s participation in this appeal...
In the haze of references to closed minds and reasonable apprehensions, one thing is clear: Croswell's support of anti-quarrying proposals was the critical factor underpinning the Court's conclusion. Applying the test in law for determining reasonable apprehension of bias, we conclude that the decision of the SDAB here meets that standard. The decision of the SDAB is quashed and the matter is returned to the SDAB for fresh proceedings consistent with these reasons. It goes without saying that Croswell should not participate further.
As an application of the closed mind standard, I think this is harsh. The fact that Croswell supported anti-quarrying ordinances did not necessarily mean that he would be incapable of persuasion as to the merits of any particular proposal. The proper question for the Court to ask itself was: Would it have been impossible to persuade him that a particular project satisfied his objections? In the absence of specific comments by Croswell, it is difficult to see how the Court could conclude that he was not open to persuasion. Indeed, the failure of Croswell's legislative endeavours meant that, as a matter of law, permits could be granted. To hold that Croswell would ignore the failure of his legislative proposal is to hold that Croswell would act in bad faith by flagrantly ignoring settled law. Such a holding surely requires significant supporting evidence.
More generally, applying the closed mind standard on the basis of prior legislative proposals is troubling. The Court seems to give no weight to the fact that Croswell was engaged in a legislative activity in proposing the ordinances. The Court's logic puts councillors like Croswell in an individious position. If he or she proposes measures which he or she believes to be in the community's best interests, he or she may be disqualified from subsequently serving on committees which grant licences which affect members of the community.
So, either Croswell convinces a majority of the council of the wisdom of his measures, or he forfeits the right to convince a majority of a decision-making committee of the merits of a particular proposal. And, one might also wonder, what of the councillors who supported Croswell? Are they too disqualified from serving on decision-making committees? Is the ultimate outcome that only those councillors favourable to quarrying can determine whether or not permits for quarrying should be granted?
It cannot be consistent with democratic decision-making at the municipal level to penalize legislative activities in this way. A better approach, in my view, would be to institute a general rule that such activities cannot ground a finding of bias, perhaps with some exceptions.
In summary, the Court's statement of the appropriate legal principles seems muddled; its application of the closed mind standard seems harsh; and the very applicability of the closed mind standard to legislative activity seems dubious.
This last aspect might tempt the Supreme Court of Canada into having a look at the case.
A quick postscript on the Facebook aspect: I suppose one could imagine circumstances in which an elected representative was a member of a Facebook group opposed to a particular development, or made his or her opposition to certain types of development clear in a permanent part of their Facebook profile. In such circumstances, perhaps the continuing nature of the elected representative's publicly expressed views would ground a finding of bias, on the basis that they were expressed -- or, because of their permanent nature, continued to be expressed -- during the hearing. But those are not the facts of this case.
UPDATE: small edit to add text from paragraph 26