Actually I think the public tends to generalize their complaints/injuries and act in the most spiteful ways that are available to them. For example, decades of bad experiences at the DMV translates into cries that we should defund the post-office, NSF, etc, no matter how irrational that is.
But capital has a playbook now that's pretty effective at dodging this kind of backlash, like the "advertising without signal" thing that's also on the front page right now is pointing out. That article mentions "Disposable brand identities" which does seem relevant here even if that piece is mainly talking about the relationship between amazon/manufacturers/consumers. Part of what PE is accomplishing is brand/liability laundering, but brands head in this direction anyway before they fail. Consumers can't typically look at list of 10-20 "different" hotel brands and really tell which are under the same umbrella.
And all this is kind of assuming consumer choice exists and is still meaningful, but when you need a hotel you need a hotel. If Hyatt gets away with this abuse, every hotel will do it soon, and capital can just wait out any boycott.