Even if QM were fundamentally deterministic, and I agree that there is experimental evidence suggesting that pre-collapse that the probabilistic description may be "real", it is necessarily at least "chaotic" in the mathematical sense that outcomes are sensitive to initial conditions so close to each other than we have no way to measure the differences between initial conditions large predict materially and measurably different outcomes.
Plenty of systems that we routinely model as stochastic (i.e. probabilistic) and random are ultimately deterministic at a level not directly influenced to a meaningful degree by quantum uncertainties: coin flips, drawing lots, rock-paper-scissors, statistical thermodynamics, weather in the medium term time horizon (several days to several weeks out), genetic mutations, the mixing of genes that happen in sexual reproduction, stock market behavior, epidemiology, etc.
So, there is nothing unusual about treating a system that is fundamentally deterministic as if it were stochastic for purposes of practical applications in a wide range of circumstances. Indeed, this reality probably (as used for this word in this sentence expressing lack of knowledge rather than a probabilistic process) has a lot to do with our psychological drive to find a deeper deterministic explanations, even when we have nothing by psychological inclinations to support the believe that QM has a deterministic basis.
Coming up with a deterministic model that accurately reproduces both pre-collapse experimental effects and entanglement effects (particularly transferred entanglement effects), probably isn't impossible, but it is very hard, at a minimum --> that is without allowing for either retro-causality, or non-local effects that have no mediator.
Retro-causality (i.e. that a cause can trace along a world line of particles that have contact interactions with each other both forward and backward in time) is probably the easiest sacrifice to make, IMHO, to reproduce entanglement phenomena in a deterministic but mathematically chaotic model of quantum mechanics, at least from a model building and conceptual perspective, without disturbing other important features of our QM models. But, it has been pretty powerfully demonstrated over the last century or so that you can't cling to all of the foundational assumptions applied to systems that are well described by classical mechanics if you want to have a model that reflects what the mathematics of QM produces in a way that is exquisitely accurate. You have to sacrifice locality or causality or "reality" (a somewhat technical term of art in this context) or determinism, or more of than one of them, if you are going to get a model that works.
This still leaves the measurement problem and how to shoehorn the observed differences between non-collapsing unmeasured systems and collapsing measured systems in a deterministic model.
I'm not aware of any very intuitively satisfying resolutions to that (i.e. to why physical things should act differently when they are measured than when they are not measured), which is why "shut up and calculate" has become kind of the default "QM interpretation", if you can call it that, even though it doesn't really express any opinion on an underlying mechanism that produces the mathematical behavior that QM tells us it should when we calculate it.
To some extent, "shut up and calculate" comes with the hidden agenda of trying to change your intuition about phenomena that are outside the non-instrumentally assisted observational experience of either us, or our evolutionary ancestors, that gave rise to our intuitions about how the universe ought to act in the first place. Calculate enough, and you'll develop an intuition about how systems where QM behavior matters behave without the "crutch" of having to associate it with one of the leading QM interpretations.