Does a Turing machine really need an infinite tape
The answer to your alternative question is "yes". A Turing machine that halts, halts in a finite number of steps, and it can only move the head a finite distance from the starting position in a finite number of steps.

However, a Turing machine need not halt and may use up an ever growing number of tape positions as it executes. Further, we cannot in general tell how much tape a Turing machine will need to execute correctly.

Sometimes people speak of needing an "unbounded" or "ever extendable" tape rather than an "infinite" tape. About the only significance to this difference in wording is that the former wording makes it a bit clearer that the notion of a Turing machine should not be problematic to a finitist (though perhaps still for an ultrafinitist).