There's a case to be made that even those who can't afford rent by their own wages aren't fully proletariat. The fact that they can still survive is due to reliance on the super-exploitation of the global south, which pushes them towards being labour aristocrats. (Albeit, this part of the transfer is wealth from south to north is under threat.)

But for the people identifying as middle class who can't pay rent, is it an ideological thing? i.e. they think they're middle class because they're parents were or because although they get low wages, they're in a 'middle class' job? I'm not sure I'd call relatively poor people middle class. But there are people who call precarious knowledge workers 'middle class' just because they don't get their hands dirty at work.

In this sense, 'better-paid workers' does not necessarily equate to 'better-off proles'. I should've been clearer that I reframed your comment. Still, this is why you're original point was fundamentally right—'middle class' is a slippery term and not nearly so useful as class concepts defined in relation to the means of production.