Does the wave-function describe a physical system or our knowledge of that system? No.

It describes neither. So what’s the use of wave-functions?

The primary use of a wave-function is to prescribe (not describe!) how firmly to believe claims about the values of physical magnitudes on a physical system.
Anyone who accepts quantum mechanics uses the wave-function to do this by plugging it into the Born rule and adjusting her degrees of belief to match the corresponding Born probabilities.

Probabilities for what? The values of magnitudes (x-position, z-spin, energy,…). So the Born rule should be stated so as to assign probabilities to claims about the values of magnitudes, not claims about measurement outcomes.
But don’t we know that can’t work, because there is no consistent non-contextual simultaneous assignment of values to all “observables”?
There isn’t, but there doesn’t need to be. The Born rule can be legitimately applied only to significant magnitude claims!

You mean claims about magnitudes in an experimental arrangement suitable for their measurement?

In general, “No”: plenty of magnitude claims about what happened long ago in a far-away uninhabited galaxy are perfectly significant. If you are concerned about which these might be, ask whether application of a model of decoherence would pick out a “pointer basis” close to diagonal in a basis of eigenfunctions of your favorite “observable” magnitude on your favorite quantum system. If yes, it’s O.K. to apply the Born rule: otherwise, not.

Why do this?

1. It is consistent: Kochen/Specker-type proofs don’t rule it out.
2. It works: beliefs formed in this way are reliably confirmed by experimental statistics: that’s why you should accept quantum mechanics and use it this way.

What about the measurement problem? What about Bell?

The measurement problem arises only if one mistakenly takes the wave-function completely to describe a system to which it is assigned, including the entangled system+apparatus in a measurement. Since it doesn’t describe a system at all, no problem! (Quantum mechanics can’t explain why there is a unique outcome since its application presupposes that there is. Is that a problem? Only for those who put unreasonable demands on quantum mechanics—it’s still the best theory around!)

Bell showed that no theory of local beables satisfying a factorizability condition he took to follow from an explication of an intuitive local causality requirement is consistent with certain quantum predictions, now amply confirmed by experiment. Quantum mechanics doesn’t satisfy this factorizability condition, but Bell’s argument for that condition from the intuitive local causality requirement makes assumptions that fail in quantum mechanics. Specifically, it assumes that an event has a unique chance. But the use of wave-functions in generating Born probabilities shows why both wave-functions and chances must be assigned relative to the space-time location of a hypothetical agent applying quantum mechanics. When this is done, quantum mechanics can be used to explain violation of Bell inequalities with no superluminal influences.