That doesn't necessarily follow - the universe can be infinite in size, but contain a finite amount of matter.

Not the universe we observe, no. There is no valid model in GR that has this property and matches our observations of the universe as a whole. Models with a finite amount of matter surrounded by an infinite region of vacuum exist in GR, but they are not homogeneous and isotropic on large scales, while our observations indicate that our universe is.

True, I was keeping the reasoning about the average density. A homogeneous universe is still the null hypothesis.