The infinite tape part isn't some minor detail, it's the source of all the difficulty. A "finite-tape Turing machine" is just a DFA.
Oh is that all? If resource bounded Kolmogorov complexity is that simple, we should have solved P vs NP by now!
I debated adding a bunch of disclaimers to that parenthetical about when the infinite tape starts to matter, but thought, nah, surely that won’t be the contention of the larger discussion point here haha
The point is that a finite tape isn't enough for an LBA.
I suggest that you read Michael Sipser's Introduction to the Theory of Computation. It's absolutely lovely and an easy read, considering the subject matter. It will help you to understand cardinality (and many other things) in a pragmatic computing science context.
> Infinite means, well infinite
There's a technical definition of "infinite cardinality" which is a bit more rigorous than "well infinite". If |S| > n for every natural number n, then S has infinite cardinality basically by definition.
Why don't you try implementing an LBA simulator in your favourite programming language with a finite-sized array and tell me how it goes? Anyway, the original point was that a Turing Machine with a finite tape is an FSM. This is true (well, or more technically: it's equivalent to an FSM) because you can encode all possible configurations of the tape with a finite set of states and thus don't need any extra memory.