Key word is provably. We can assume the universe is finite because we don't have evidence against it, but no one knows if it's possible to prove or reject that for good.

Infinity is an abstraction. The universe cannot be "infinite" in the mathematical sense, as the mathematical infinity is an abstraction. The only meaningful question is "is mathematical infinity required to produce good models of the universe/physics?" and the answer so far is "no". We use infinity because it is convenient, but it is not really needed.

There is no reason to think that "infinity" will be essential to describing the universe any more than any other possible abstraction that we have not so far constructed.

Why can the universe not be infinite and go absolutely indefinitely in every direction?

> The universe cannot be "infinite" in the mathematical sense

Why not? An infinitely large universe goes against our intuitions, but our intuitions were formed by the experience of living on a finite planet and have no bearing on what's true on a galactic scale – just like they have no bearing on what's true at the quantum scale. There is no objective reason to assume that the universe isn't infinite.

Observable universe is just the part of the universe from where light had time to hit the Earth already. It is growing pretty much at the speed of light, not because of physical changes to the space-time, but because the light from farther away finally arrived. So while it's definitely finite, I wouldn't say it disproves the claim that nothing about the universe is known to be finite.

> It is growing pretty much at the speed of light

The observable universe is growing much faster than that, because space itself is expanding.

Because of inflation, there is a distance at which the objects not already in our light cone are moving away faster than the speed of light. So the amount of stuff that we will be able to observe is limited.

besides the other points raised, even within the observable universe there are some assumed infinities as well as things that might be infinite, might not be.

e.g. time and space likely aren't quantized, you have an infinite amount of points in between any two arbitrary points in spacetime. don't mention planck scale black holes, it's a meme and even most phycisists who like to cite it haven't actually thought about it for more than 5 minutes.