No, I understand this point. And agree that compelled speech is a very dangerous principle to play around with. But the question isn't whether you should be compelled to say something you don't believe (you shouldn't), but whether you should be able to deny service to somebody whose protected characteristics you disapprove of.
It's one thing if somebody is asking for a cake that says "Homosexuality is God's natural order." But another if it's simply saying "Happy birthday, Dave. Love from your husband." I don't think a business should be compelled to write the first cake, because it expresses an opinion they may well not believe. But refusing the second is discrimination based purely on the protected characteristic of sexuality (in fact, ridiculously enough, it's based on the name. You could leave everything the same but change the name to "Davina" and it would be fine).
To give another example, after segregation, companies were compelled to offer services to black people. They didn't have to like it. They weren't compelled to say they were happy about it, but they were compelled to offer their services, just as they would be to anybody else.
Of course, black people would much rather not be served by bigots. But the principle has to be that they are granted the same access to services as anybody else. And without compulsion, that wouldn't be the case.
But to your first point, I'm not trying to de-legitimise the court. My question is simply, what is the role of the court? And is it performing that role? Is the way it's currently set up the best way to perform that role? I'm far from the first person to ask these questions. But I think they're important. And I don't think you can put rulings (or actions and comments) from people like Alito and Thomas aside when tackling those questions. Some of Thomas' conduct and comments have been disgraceful.