Is it possible to conceive of a universe macroscopically similar to ours in which matter is NOT fundamentally composed of oscillating waves, or would any such universe be logically contradictory?

It is ultimately a philosophical choice not demanded by the mathematics to actually interpret reality as oscillating waves. Erwin Schrodinger for example argued against the notion that particles really "spread out" as waves and instead argued that the particle just kind of hops from interaction to interaction without having meaningful existence in between interaction. If you go this route, then the wave function doesn't "describe" anything, but rather predicts where particles would hop to during an interaction.

The reason Schrodinger argued in favor of this is because he said treating particles as actually spreading out as waves contradicts with the fact we only measure particles, so you need an additional postulate that says these waves suddenly collapse back into particles the moment you try to measure them, and he did not see why "measurement" should play a fundamental role in the theory. This is sometimes called the "measurement problem" and Heisenberg's formulation and interpretation does not have this problem.

If you mean, can you get rid of the wave function entirely, the answer is also yes. When quantum mechanics was first formulated, it was formulated using Heisenberg's matrix mechanics, which make all the same predictions but does not use the wave function. The wave function is a result of a particular mathematical formalism. There is another formulation of quantum mechanics called the path integral formulation, and yet another called the ensemble in state space formulation.

The probability of finding an electron or any other particle at one point or another can be imagined as a diffuse cloud, denser where the probability of seeing the particle is stronger. Sometimes it is useful to visualize this cloud as if it were a real thing. For instance, the cloud that represents an electron around its nucleus indicates where it is more likely that the electron appears if we look at it. Perhaps you encountered them at school: these are the atomic ‘orbitals’.

This cloud is described by a mathematical object called wave function.The Austrian physicist Erwin Schrödinger has written an equation describing its evolution in time. Quantum mechanics is often mistakenly identified with this equation. Schrödinger had hopes that the ‘wave’ could be used to explain the oddities of quantum theory: from those of the sea to electromagnetic ones, waves are something we understand well. Even today, some physicists try to understand quantum mechanics by thinking that reality is the Schrödinger wave.

But Heisenberg and Dirac understood at once that this would not do. To view Schrödinger’s wave as something real is to give it too much weight – it doesn’t help us to understand the theory; on the contrary, it leads to greater confusion. Except for special cases, the Schrödinger wave is not in physical space, and this divests it of all its intuitive character. But the main reason why Schrödinger’s wave is a bad image of reality is the fact that, when a particle collides with something else, it is always at a point: it is never spread out in space like a wave. If we conceive an electron as a wave, we get in trouble explaining how this wave instantly concentrates to a point at each collision.

Schrödinger’s wave is not a useful representation of reality: it is an aid to calculation which permits us to predict with some degree of precision where the electron will reappear. The reality of the electron is not a wave: it is how it manifests itself in interactions, like the man who appeared in the pools of lamplight while the young Heisenberg wandered pensively in the Copenhagen night.

--- Carlo Rovelli, "Reality is Not what it Seems"

Of course, you might say that this is still not "macroscopically similar to ours" because in our classical world we do not need to treat objects as if they only exist in the moment of interaction. There is always a tradeoff in quantum mechanics. It's not a classical theory. There will always be some differences, so it really depends upon what differences you find the most intuitive/acceptable. If you find the oscillating wave picture to be too bizarre then you can think of them just as particles, with the tradeoff that they only exist relative to what they are interacting with in the moment.

It would be pretty hard. There’s a reason quantum mechanics is the current explanation, and it doesn’t start with the Bell entanglement experiments.

Black body radiation would have some bizarre behavior without quantum mechanics.

The radiation spectrums of stars are also very dependent on quantum mechanics.

Some related phenomena such as transistors and phosphorescence are hard to explain without quantum mechanics.

A big one is chemistry is highly dependent on quantum mechanics. You could have a limited understanding of ionic compounds with just the Columbic force, but covalent bonds require quantum mechanics to explain.

Most of physics history is studying the edge cases and gaps in the current understanding, and filling those in. Quantum mechanics didn’t just appear suddenly; it was derived as an explanation for many previously unexplained phenomena in pieces my many different people over time.

Sure. It could be an abstraction based on physics in a higher universe to which there are impossible barriers to accessing further understanding.

Just look at any video game universe.

Sure, if you're making up all the rules you can make up all the rules. Matter could be composed of the body of a dead god, for example.