Your argument isn't clear to me, but with the cake message example, I don't think it does involve to "visibly align yourself with their values".
My argument was simply that discrimination on the basis of someone's beliefs is not the same as refusal to accept the truth or rightness of those beliefs. It is certainly a rather clumsy phrase - I was attempting to address the fact that you seem to struggle with the distinction between (on the one hand) providing a service to someone with whom you disagree, and (on the other) pretending you agree with them. I'm not sure how I can say that more plainly. Would you find "express views you disagree with" easier to understand? If it helps, here is the relevant piece of the summary of the Supreme Court judgement, which is probably about as clear as anyone can be:
"The rights to freedom of thought, conscience and religion (article 9) and to freedom of expression (article 10) were clearly engaged by this case [49]. They include the right not to be obliged to manifest beliefs one does not hold [52]. The McArthurs could not refuse to provide their products to Mr Lee because he was a gay man or because he supported gay marriage, but that was different from obliging them to supply a cake iced with a message with which they profoundly disagreed [55]."
Except, I suppose, insofar as if you sell a guitar to a trendy vicar you are "forced to contribute" to Shine, Jesus, Shine. Or if you rent a flat to a Catholic couple you are "forced to contribute" to them indoctrinating their children under your roof.
Again, not sure what you are arguing here.
As I understand it, your argument was that providing a service to someone (e.g. allowing them to meet in your space) might enable them to promote their beliefs, and that therefore you were "forced to contribute" to the furtherance of their (objectionable) views, even if you did not express those views directly yourself. Therefore, you appeared to be saying, it was an infringement on your right to freedom of speech equivalent to a demand that you express those views as if they were your own. I was pointing out the logical implication of this argument, i.e. that if it were so, you would be justified in refusing service in many situations which at the moment would clearly be illegal.
Essentially you seem to be arguing for the removal of "religion or belief" as a protected characteristic, which is not in itself an incoherent position (although it is a totalitarian one). As DemiColon's excellent post points out, there are balances and compromises to be made, and for some people the lines have been drawn in the wrong place. (Hence, for example, the need for the Grainger criteria, and the arguments about whether beliefs are Worthy of Respect in a Democratic Society, etc.) But if so, you have to live with signs in shops that say "No Jews" or "No Muslims" or indeed "No feminists". Are you happy with that?