Standard proof is a diagonalization, which depends on infinite sequences.

That said, the Halting problem on physical computers is still of extremely high computational complexity, which can be just as high a barrier as incomputability.

Of course it does (implicitly!).

Otherwise the number of possible states of the Turing machine T that we want to check whether it halts or not is finite and we just have to "simulate" the program and see whether it terminates or it reaches again a state that it was already in.

How would the "if this program halts then run forever, else halt" algorithm work on a real computer? I imagine it would fail with stack overflow.