It's not that the motifivation / generating idea is definitely wrong, the problem you're alluding to here is just that they form an asymptotic series[1]. An asymptotic series is one that diverges if you sum the whole series, but will accurately approximate some function if you take a finite partial sum.
The reason that so many perturbative series are only asymptotic in quantum mechanics actually makes a lot of sense if you think about what these perturbative series are saying. Typically, each order of the perturbative series is meant to account for the physics at smaller and smaller length scales, but we know that our physical theories aren't applicable at infinitely small length scales because we know that at the very least, we need a theory of quantum gravity.
So in some sense, the problem is just that we shouldn't even want to sum these series to infinite order because there is no reason to think that the ultraviolet behaviour in that series has any physical meaning.
2. Those ideas can't possibly be correct, because quantum gravity can't quite fit in.
3. Suppose anyway that the ideas are good enough; look at this cool equation.
4. Yeah, it doesn't... technically...converge, but the first few terms are pretty good.
Your comment about not knowing things in the limit yet makes perfect sense, but the motivating argument for that math in the first place is limiting behavior. It makes a ton of sense that none of these ideas converge, but it's peculiar that some of them are basically right for small terms when they're created from a foundation of convergence.
> but the motivating argument for that math in the first place is limiting behavior.
No it isn't. The entire reason we truncate the series and dont sum the entire series off to infinity is exactly that the limiting behaviour isn't well defined.
The reason that so many perturbative series are only asymptotic in quantum mechanics actually makes a lot of sense if you think about what these perturbative series are saying. Typically, each order of the perturbative series is meant to account for the physics at smaller and smaller length scales, but we know that our physical theories aren't applicable at infinitely small length scales because we know that at the very least, we need a theory of quantum gravity.
So in some sense, the problem is just that we shouldn't even want to sum these series to infinite order because there is no reason to think that the ultraviolet behaviour in that series has any physical meaning.
[1] https://en.wikipedia.org/wiki/Asymptotic_expansion