No, they'll simply deny they ever held the view in a few years, all on their own. See: Segregation.

Why might they do that, theoretically speaking?

Edit: A downvote for asking a question? Really? Is the answer that terrifying?

Hopefully because they'll no longer be able to justify - even to themselves - why they thought those views made sense in the first place.

Because they'll be socially shunned by people who find their views to be embarrassing.

Is shame, then, the way social issues should be addressed?

Separating this sort of stuff into "social issues" is I think an incorrect way to look at it.

We're talking about changing normative group positions in a complex society that has delegated huge swaths of norm policing to immensely powerful institutions.

So, shame is one way. You have to make it shameful in the first place - and then you still have to change the institutions.