Hilbert uses physical arguments to deny the existence of the infinite anywhere except in thought. Certainly any convincing account of the relationship between the infinite and the real world would have to be fairly subtle. For this reason, finitistic mathematics has always been much less controversial than infinitistic mathematics. Only in our own time has there arisen an ultrafinitist school which posits bounds on the length of the natural number sequence. And the ultrafinitists have neither refuted finitistic mathematics nor shown us what an ultrafinitist textbook would look like.
Finitistic mathematics is as firmly grounded as a science can be.
The essence of Hilbert's Program is to reduce infinitistic mathematics to finitistic mathematics. I grant that the reduction of infinitistic proofs to finitistic ones does not increase confidence in the formal correctness of infinitistic proofs. What such a reduction does accomplish is to show that finitistically meaningful end-formulas of infinitistic proofs are true in the real world. Hence formulas which occur in infinitistic proofs become more reliable in that they are seen to correspond with reality. Finitism is much more restricted than either predicativism or intuitionism.