I mean, does this sound like a thing that would make people want to be nicer to groups that value history and/or tradition, especially traditional social roles? It doesn't to me.

It sounds to me like a country that is doing a fairly good job of ensuring freedom of conscience and freedom of expression. The question is specifically not whether one group agrees with another group, but whether those groups have the freedom to act as they choose, so long as they are not infringing on anyone's civil rights.

Infringing on someone's civil rights would be something like forced marriage; that's not about to be okay, as it constrains someone's freedom of conscience (the person being forced to marry). OTOH there is no valid claim to be made that, for example, a gay couple not being allowed to be married in the temple are having their civil rights infringed. There is no right to being admitted to a private organization, and certainly not to a religious one; this has been upheld over and over and over again by the Supreme Court. Thus, you cannot force a rabbi, minister, LDS bishop, or any other religious figure to marry you within their religious tradition; no one has that right, as it violates the freedom of expression and conscience of others in the religion.

It sounds like they'll say, "look at all those terrible traditions we don't tolerate anymore! We're not going to tolerate yours anymore either. Comply or die." Which is frankly exactly what they're being programmed to say.

Specific examples, please?

I don't believe we'll be left alone regarding the Church's policies on homosexuality, I think that pressure will come to treat homosexuals as full members, including temple admittance, or face the wrath of the civil authority.

Well again, you'd need something beyond your opinion to give that any heft. There's plenty of evidence against it in how laws and court rulings have been made regarding private and religious organizations, and none that I know of that give me pause in this regard.

You may believe things without any evidence of course, but I don't believe it's wise to give into negative beliefs that spawn fear or apprehension without significant reason to do so. You may disagree of course.