The case is that no scientific experiment or computation is or ever will be infinite in size, energy, time, or repeatability.
There is no other way to realize infinity physically. This is a consequence of spacetime and QM and our finiteness.
Even if the universe is infinite, we cannot seemingly prove so scientifically.
Is infinity in math required for science? There are other mathematics and formalisms that do not use it. Since all empirical data are finite, these formalisms may be sufficient, if more time consuming to work with.
But even arithmetic uses principles of infinity, and so does set theory. One would have to work outside those, but in principle one probably could.
Besides strict finitists are fictionalists like Field. They do not necessarily take mathematical statements as true, at least true as in how you’d describe physical objects. Truth as in Sherlock Holmes has a pipe is true.
Given that infinity is omnipresent in modern math and science, because it is useful, even the finitist must recon with it.
Thus disbelief in infinity requires fictionalizing large parts of mathematics essentially. And yet doing so does not explain how relying on infinity can produce effective mathematics even when it may or doesn’t exist physically or platonically.
Infinity is present in our most basic mathematics (arithmetic) and is foundational (set theory, etc). Infinity has provided a paradise of effective math and theories, and without it theorizing becomes more difficult. Removing infinity from mathematics has failed since Cantor but we still don’t know what mathematics is so mathematical infinity cannot be leveraged to say it actually exists beyond the mind. It may be no more real than a fairy tale. Yet it is no doubt useful. And other finite formalisms at least could be utilized by science, but that doesn’t get around why infinity has been useful.
It will never be empirically proven to exist. But that doesn’t make a strong case against it. Other things won’t won’t be empirically proven which we believe exist (stars outside the observable universe say). What makes a stronger case is that there seem to be capable finite mathematical alternatives, and statements can be fictional but still useful.
Of course Cantor, Godel, Badiou, etc would scoff at this. Infinity is more real than the finite to Badiou. But the case has been made from the other side like Field who is ready to levy some harsh criticisms of modern math.
But I don’t think you’ll get “incoherence” of infinity even from the doubters.