1) A quantum state can exist in a superposition of different states, but once measured, it collapses into one and only one state. If measured again it will still be in that state. So it is "concrete" in that sense.
2) "Contextual" is a word usually used to describe language, not physical measurements. However, if you mean that the state of a system depends on the state of every other system around it, then that, too, is not correct. Special relativity and the speed-of-light speed limit place hard constraints on what can affect an event at a specified time and place.
On your (1), the typical interpretations for QM say that the wavefunction is not ontic, and this is directly traceable to the Copenhagen school of thought. QM is best done when we consider that our knowledge of systems is always partial; it turns out that often, the system itself is also partial about knowledge of own state. I greatly enjoyed reading [0] recently for a coverage of a toy theory exploring this distinction in more detail.
On your (2), quantum contextuality [1] means something more subtle than your phrasing; it means that the state of a system is not the entirety of what is measured by an observer. To use your phrasing and state something very strong, the Free Will Theorem [2] says that the state of a system informs, but can never totally control, the way that a particle chooses to reply to an observer taking a measurement. This both opens the door for new avenues in free-will philosophy while slamming shut the idea that e.g. determinism can be meaningful or possible.
[0] https://arxiv.org/abs/quant-ph/0401052
I would just point out that,
a) it's not observer dependent, it's measurement dependent. If you collapse a wave-function in a superposition of states into state A, when I measure it, it will be state A as well. In that sense it is very concrete.
b) surely you can see my confusion at your statement on contextual vs. quantum contextuality. For latter is a technical term, but the former was used in a non-technical way.
Whatever meaning you're using, there is something called "Tarski's theory of truth", it uses formal tactics extensively, and describes how to do formal semantics for a language.
Edit: https://www.logicmatters.net/resources/pdfs/godelbook/GodelB... is quite good and includes Tarski's results. There's also GEB, of course.
Tarski's work described a criterion for a theory of truth for a language, and showed how to give a compositional semantics for some formal languages. As you noted, that language does not define its own truth predicate, to avoid paradox, but this is a formal analysis of truth. His work, along with Gödel's, ended the logicist program, but not formal treatments of semantic concepts.
P.S. Don't assume what others have or haven't read. I can't claim any expertise, but I took a graduate seminar on the theory of truth taught by Anil Gupta. That doesn't mean I'm right, of course.
To be blunt and opinionated, I see the 1890s as asking the questions which we would answer in the 1930s. We found that set theory has gaping holes, and the answer turned out to be that all maths has gaping holes. We found that classical mechanics can't describe ovens, and it turns out that we need QM for describing lots of low-level physical phenomena.
Logic is often classed as a branch of philosophy, and that covers the Gödel work, and more, mentioned.
There is plenty of writing, much of it by real philosophers, about how foundational work in QM, for example, did have profound philosophical implications and did influence the development of philosophy.
The physicists who invented QM had many discussions about its philosophical implications, and these discussions have been studied by philosophers and influence philosophy.
The division separating “philosophy” from mathematics, physics, and even logic is recent and arbitrary.
And finally, if an academic philosopher finds math too hard, that is not a good reason to disqualify it from being considered “philosophy”.
Of the logicians/computational theorists I mentioned, three have their own entries [1][2][3] and are each associated with their own philosophical bodies of work. Von Neumann lacks his own page, but makes up for it by being a recurring secondary character in stories of Russell and Gödel.
The QM physicists fare similarly, but are not credited as Great Men. I am okay with this. Bohr is discussed alongside his Copenhagen school [4], Heisenberg alongside his Uncertainty Principle [5]; but Pauli does not get a full page, being split between [4] and [6], a second Bohr page.
While I haven't shown that these folks are central to philosophy, I think that I've shown that some philosophers definitely consider them to be directly influential upon some part of the practice.
[0] https://plato.stanford.edu/
[1] https://plato.stanford.edu/entries/goedel/
[2] https://plato.stanford.edu/entries/turing/
[3] https://plato.stanford.edu/entries/tarski/
[4] https://plato.stanford.edu/entries/qm-copenhagen/