Should you review as most reviewers would? Or should you follow what you think is scientifically correct?
I'm not convinced this is a binary choice as you describe it. A good review probably isn't either of these.
Your review is sought, to check the papers adequacy and caliber. As part of that, if they wanted "most" reviewers views, the editors are more than able to ask other more typical reviewers, or a wider range of reviewers, to review it. Therefore you should assume they actually want your view, not a mere echo of what you perceive to be a popular view.
But giving your view need not be idiosyncratic or fringe-y either.
If the field's methodology is weak or flawed, there are, presumably, reputable papers that say this. You can allude to those papers, and state that you have concerns that the paper is prone to/may have suffered from the weaknesses X, Y and Z, as described in [list of cites], because [reasons]. You ask the authors to address these concerns as usual.
If there are no such papers, or they are not seen as significant in the field (or overlooked), then describe that you see a possible concern that the algorithm may be so tuned to the specific data, that it is unclear if it has general interest, because the authors do not appear to show its performance against appropriate general data. (And if needed: the data they do test against cannot in your view be considered appropriate as a test with general data, because (reasons), notwithstanding that it is a widely used dataset). That too, is a sensible, professional statement.
Really, your review is to state what, in your view, needs to be addressed, in order that the paper become acceptable, adequate, and professionally worthwhile publishing (if not already so). So you are not advocating a view, so much as identifying possible inadequacies (as you personally yourself feel they may exist), that you convey to the authors so they can address them - subject to the editors overriding judgement on the need for this and willingness to publish.
Because you are recounting possible issues to check, rather than advocating a position, you should find that there is a wording that allows you to state you have a concern, without adopting a fringe-y position in doing so.