There is no practical difference between denying and assuming infinity since we are unable to carry out any practical experiment that would tell us which of the two options is true.
More generally, falsehoods don't make any difference to any practical use as long as nobody in fact uses them.
This is indeed routinely applied in everything we do including science. False theories are regarded as perfectly scientific as long as nobody finds a way of putting any falsehood in them to any practical use. When somebody does, the theory of course is falsified and scientists work to think up another one which would be without at least this particular falsehood. This seems to have worked well so far.
Human beings seem to be finite at least in the quantity of information they can handle and the speed at which they can compute solutions. The logic of the human mind not only is organised to accommodate this fact but it is indeed itself both the result of, and optimal solution to, this fundamental constraint.
Denying or asserting the reality of the infinite are both self-indulgent metaphysical views. Assuming the reality of the infinite should be good enough. It will always work since no one will ever be contradicted by the facts of the matter.
Claiming infinity doesn't exist is just as futile but you also won't risk being contradicted by the real world as long as you refrain from specifying the boundary of the universe.
The only sort of claims that would be falsifiable is for people who believe the universe is finite to make: They could assert what the limit is. Strangely enough, many people claim infinity doesn't exist, but very few of them want to assert what the limit would be.
Many people in the past, broadly before Copernicus and Galileo, would confidently assert that the universe didn't extend beyond "the outermost sphere of the heavens", that is, broadly, the "firmament", or celestial spheres of the planets and stars.
This claim of course was falsified, several times over as people realised, in stages, that the world extended beyond all visible stars, beyond Andromeda, the only galaxy visible, if barely, in the night sky, with the naked eye, and which is already a staggering 2.5 million light-years away from us, and indeed extends well beyond whatever anyone could possibly imagine in that narrow, indeed limited, space of the mind which is available for people to do their imagining in.
Thus, reasonable people have learnt from experience to limit their claims, if not their intimate conviction. They choose to merely assert that the universe is finite without specifying what the limit might be, and thus without risking being contradicted by the fact of the matter.
Perhaps the fundamental question regarding our notion of the infinite is how come, with our very limited brain and very limited mind, we nonetheless came to conceive of the infinite.