The concept of a modal interpretation is even vaguer in its implications than many-worlds and retrocausal interpretations. The only unifying concept seems to be that other worlds exist in the discourse, but they are purely counterfactual and play no role in explaining anything that happens in the one actual world.

I don't agree that the concept of a modal interpretation is vague. The basic concept is that a quantum system can have a physical property with a definite value without its wavefunction necessarily having to be in an eigenstate of the corresponding operator. So the eigenstate-eigenvalue link is not bidirectional. That's basically it.

The only vagueness is that there are then multiple interpretations, each of which assigns different properties as the ones which have the definite values. So which properties does the system have then, and how can we tell?

There are various theorems ("Hardy theorems") proving that an ontological theory can't assign definite values to all observables in all quantum states, such that all QM expectations are satisfied simultaneously.

I think you mean Kochen-Specker theorem here (and similar results going back to Gleason's theorem)? The system can't have definite (non-contextual) values of all operators at once, because the operators don't commute. Particular interpretations build a maximal set of properties which the system can have at once. Hardy's theorem seems to be related to whether the properties can be Lorentz invariant or not.

As you say, Bohmian mechanics is one of the interpretations (based on assigning definite position states to all particles at all times), but perhaps is not the most plausible one, since the choice of the position operator as the "preferred" operator is "put in" by hand. Other interpretations try to allow the preferred operators to "drop out" of the wave function (via measurement events or other decoherence events) rather than being "put in" and are in that sense simpler (fewer assumptions) and more plausible.