You may be assigning too much meaning to the term "field".

All that a field is in physics is a physical quantity that has a value at every point in spacetime.

So if we observe an electron, and we believe that electrons can appear anywhere in spacetime, then we define a field with the relevant properties, as an abstraction that allows us to model electron activity in spacetime. Aside from the mathematical details of the representation, that's all there is to it.

(Clarification: "anywhere in spacetime" can actually be a subset of spacetime, e.g. many particles could not have appeared in the early universe before the electroweak phase transition. We say the universe then had a different vacuum structure, which is equivalent to saying it had a different set of fields.)

This seems to me to negate your objection about "so general that...", at least for particles we observe. We define the fields to match the observed properties, and that's our model of that type of particle.

The second issue I see with what you're saying seems to be a kind of conflation of established theories supported by evidence, with theoretical developments in progress. Many scientists would agree that gravitons may not be the right approach - particularly those working on alternate theories, such as emergent gravity, AdS/CFT correspondence, loop quantum gravity, string theory, etc. (The full list is much longer!)

Those people working on the theories of quantum gravity are of course going to talk about gravitons, but they can't claim we know gravitons exist because we can't observe them and there isn't even a complete and consistent theory that describes them. That's still being worked on. But it's certainly an obvious avenue of research.

The same thing goes for what you said about the "constant tendency to quantize everything". For quantum objects we observe, we observe that they're quantum so there shouldn't be any controversy there. For possible objects we haven't yet observed, like dark matter or gravitons, exploring the possibility that they're quantum just makes sense, if the behavior of what we're looking for is consistent with that. It doesn't prevent research on other possibilities.

> So when we have a problem like dark matter, to say “maybe it’s a particle in its own field” can feel a bit like physicists going back to the same old well again.

If we're looking for missing mass, there's already a whole theory of how mass works, which is QFT. The theory predicts that anything with mass must be a quantum particle. Of course the theory could be incomplete, but it wouldn't make sense not to explore the possibility that other massive particles could explain our observations.

Besides, everything in the physical universe we've ever observed fits under either QFT or GR. As far as we know, that's how the universe works. It's natural to explore new phenomena from that perspective.

> (the fine structure constant doesn’t have an understood “cause”, for example.)

Funnily enough what was being discussed in this subthread can completely explain this. Imagine a very large, if not infinite number of quantum fields, each coupled to others in all sorts of possible ways. In that case, the fine structure constant we observe can be explained by the weak anthropic principle: somewhere in that large possibility space, there are likely to be fields with properties that can support the existence of observers like us.

The idea of "hidden" quantum fields is known as hidden sectors: https://en.wikipedia.org/wiki/Hidden_sector . According to the Copernican principle, we should take it seriously. There's no real reason to think that the particular set of quantum fields we're able to interact with are the only ones, just as it turned out we didn't live on the only planet, or in the only solar system, or in the only galaxy. (Although, the nature of gravity's interaction with quantum objects could constrain the possibilities here.)