Hasen j hacker-not-engineer. I suspect the reason is that causality in the classical sense hasn't been found to be useful enough for Physics and Engineering to justify the extra baggage. It's enough for practical purposes to construct good models for the statistics of experimental data, and the most effective mathematics for doing this is the mathematics of Hilbert spaces.
Interpretations of quantum theory exist —such as the de Broglie-Bohm and the Nelson stochastic interpretation, but there are various others— that will more-or-less fulfil what looks rather like a wish for classical causality in your Question, but they add a relatively awkward layer of mathematics that doesn't, for most Physicists, add enough insight or ability to do better Physics, Mathematics, or Engineering. There are other detailed reasons why not many physicists use these interpretations, particularly concerning Special Relativity.
We can model an experiment in a way that is as close to classical as we feel like having, while staying in the quantum theory fold, by introducing increasingly detailed models of every part of an experimental apparatus, instead of just modeling small numbers of electrons or photons, etc. This is, very loosely, called "moving the Heisenberg cut". This is quantum theory's version of "hidden variables"; model refinement happens, it's just not done by introducing classical hidden variables, it's done by introducing quantum mechanical variables that weren't in the less refined models. Consequently, there's no strong need to introduce classical hidden variables, even though it's not impossible to do so. This is loosely tied to a currently popular interpretation of what happens in quantum mechanical measurements, decoherence, which says that classical properties emerge because of the huge numbers of degrees of freedom that surround quantum systems.
You should understand that I've insinuated some unconventional views in this presentation of the conventional view. Also, there are more people in the Physics community than you might think who have committed their whole working lives trying to find good reasons why the Physics community should do Physics more in a way that, crudely, Einstein would approve of, so far without much success. New approaches and arguments emerge fairly regularly, are considered seriously by people who would jump if they could see practical advantage in it, and are found not good enough. The de Broglie-Bohm approach is essentially a product of the 50s, the Nelson approach is essentially a product of the 60s, from the 80s we have the GRW approach (apologies to anyone whose favorites I've missed out); it's more difficult to give a name to an approach from the 90s and 00s that people might cite in 20 years time, but there are many candidates, however none of them is obviously more useful.
EDIT: Try looking at all the interpretations listed on the Wikipedia page https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics, and think whether there's a way in which any of them helps to make the construction of models more practical. The Wikipedia presentations are not necessarily the best available, but the best presentations are probably not in a different ballpark as far as the level of complexity involved is concerned. It's OK to be annoyed that there's not anything better available, but constructing something better is real hard. Not a lot different from the situation for large-scale software, perhaps, where the balance of features and simplicity is also hard to master. Wanna displace Microsoft? We have to go ahead and try.
EDIT(2): In response to your EDIT,
If a particle has 30% chance of being "here", and 70% chance of being
"there", then I would assume that there's some underlying reason that
determines where the particle is. But the prevalent view (as I understand
it) is that there's no underlying reason, the particle just happens to
sort of "choose" to be "here" 30% of the time and "there" 70% of the time
with no particular reason.
As I see it, this introduces complications, because what the conventional view is depends on whether you put a question in terms of particles or in terms of quantum fields. Consider this restatement, which changes the single reason that you invoke to cause events, a particle, into a confluence of multiple or a continuum of reasons, which is much more appropriate to a field,
If there is a 30% chance of observing an event "here", and a 70% chance of
observing an event "there", then I would assume that there are underlying
reasons that determine whether we observe an event "here" or "there".
There is rather too little interpretation of quantum field theory, but it's to QFT that Physicists retreat when people press for details. The state of the quantum field describes where we can expect to see events (OK, to stay in the conventional I should say particles, so you can stop reading now, but it's fairly widely understood that the concept of a particle in QFT is theoretically problematic, whereas statistics of recorded events are measured) when we use a particular experimental apparatus. Now, if you want a reason why events happen "here" or "there", you can, if you're careful to understand that a quantum field is an operator-valued distribution, not a classical field, say that the statistics we observe are because of the quantum field, which is everywhere between the preparation apparatus and the events in the measurement apparatus. Art Hobson is the only person I know who has published on this, although some of my papers falteringly hint towards this kind of approach. Art's papers are not perfect, but they're available on his web-site, try "Teaching quantum physics without paradoxes" and try others if you like that. Brigitte Falkenburg's book, Particle Metaphysics may be too much Philosophy for most people, but I find it a very good counterpoint to Art Hobson.
This says only that the statistics of the events are caused by the quantum field. I'm ambivalent about this, I'd prefer to say only that the quantum field describes or models the statistics of the events, but you do what you like. If you want to know what causes individual events, then I'm going to leave you on your own. I think something more satisfactory than deBB-type trajectories for fields can be done in the field context, but I haven't done it. Actually, I've been trolling around here at PhysicsSE for a week or so while I recoup my spirits for a new thrash at issues that halted me a few weeks ago (I've been 20 years at it, so don't hold your breath). My approach is certainly not conventional, and you can find so many other approaches out there that there is no reason at all to read what I have to say about it. If we stay with conventional Physics, however, as you see in some of the other Answers, we can hardly engage with your Question at all.