Bath was still popular in Austen's day; it just didn't have the cache it used to.
Austen was well-aware of Bath's reputation and uses it cleverly in her writing. The Bath of Persuasion is a playground for elderly gentry, including gentlemen like Sir Walter who can't quite afford London (Mr. Shepherd is correctly appalled at the idea of sending Sir Walter off to live in London, where the baronet would bury himself in more and more debt just to keep up with the "Joneses").
From a literary standpoint, Bath is an excellent setting to show-case Elizabeth and Sir Walter's personalities. First, in Bath, father and daughter are big fish or, at least, bigger fish than they would be in London. Also, in Bath, father and daughter are able to exercise their pointless snobbery to the nth degree.
Prior to Austen's time period, private parties were considered a big no-no in Bath. People like Beau Nash went out of their way to create a society that was surprisingly egalitarian (for the time period) while also less surprisingly rigid in terms of social expectations. Going to Bath was rather like going to a really assertive summer camp where you would be expected to attend dances and concerts while getting along just swimmingly with your neighbors.
But during Austen's time period, this rigid, community spirit was waning (for one thing, Beau Nash was dead). Consequently, visitors like Elizabeth and Sir Walter are able to do whatever they please.
And what they please is exactly what you would expect from people like them:
Cocktail parties.
What Elizabeth and Sir Walter enjoy is walking around the equivalent of a ritzy hotel lobby (the Pump Room), then holding private parties where conversation is the kind of stuff you find on Twitter.
It is no surprise that Anne (and Austen) prefer the theater to this type of "entertainment." But for egoists like Elizabeth and Sir Walter nothing could be better than seeing, being seen, and showing off.
Mrs. Clay and Mr. Elliot, the heroes of my tribute Persuadable through Peaks Island Press, demonstrate no particular preference, entertainment-wise, in Austen's text. This is entirely in keeping with their goals; in both cases, they are trying to satisfy their marks--Elizabeth and Sir Walter--by telling their marks what they want to hear.