Significance of the Halting problem on non-finite inputs

Question

The Halting problem is decidable on machines with finite memory. One can enumerate all the states and continually check for a repeated state, albeit being inefficient. (https://en.wikipedia.org/wiki/Halting_problem#Common_pitfalls)

What I struggle to understand is the significance of machines with infinite memory. Why would it be necessary to model a computer having infinite memory? Any program that terminates must have finite memory.

I acknowledge that finding at which point to set the finite bar is a challenge, but I am not convinced that this task is impossible. For instance, for a program with a binary encoding of size $n$, set an upper limit of memory to $2^n$, which is sufficient to encode any possible information contained therein.

Answer 1

For instance, for a program with a binary encoding of size $n$, set an upper limit of memory to $2^n$, which is sufficient to encode any possible information contained therein.

Really? Consider the program which on input $n$ searches through $\mathbb{N}$ for $n$ distinct twin prime pairs, and halts as soon as it finds that many (and obviously runs without ever halting otherwise). Why does that memory bound work here?

The point is that we're considering unbounded searches, and even if an unbounded search halts we don't necessarily know ahead of time how long it will take. Attempts to impose a bound "right away" will lead to searches we care about yielding the wrong answer.

And beyond that, we're generally more interested in the behavior of a program in general than in any of its specific instances - with respect to the program above, we'd love to know if each of its instances halts, but no particular instance is very interesting.