The big problem with infinite speeds is that, somewhat like superdeterminism, they mean that you can't do fully controlled experiments. If effects can propagate at infinite speeds, the whe universe has an impact on any experiment, including the state of your measuring apparatus and so on. That doesn't make them impossible, but it explains why they are disliked in theories.

\* even with Newtonian physics, the universal attraction of any object does have non-0 values everywhere, but we know that the influence is negligible. However, with global hidden variables, the speed a billiard ball will take when I hit it may depend on the size of a planet in a different galactic cluster.

Every experiment does depend on the entire state of the universe, even in QM, but those influences are typically small due to symmetries. At the quantum level, many of these symmetries no longer apply.

In any case, quantum field theories are good at predicting stuff but almost certainly not descriptions of the true fundamental dynamics of the universe, given their known and relatively well understood divergences.