Oh no, not at all. My point is that the concept of infinity is in a sense necessary for the mathematics involved in developing algorithms to solve problems. We are performing induction to "predict" the behavior of an infinite number of Turing machines without actually running them. We can't just iterate through all possible programs, so we have to use patterns that apply in a consistent way to all possible problem instances to narrow down the search space.
> approximate, eventual, steady state behavior for sufficiently large N, larger than any specific one-off gimmick you might think of
I know this is how computational complexity theory is considered from the perspective of many software engineers, but my point is a bit more fundamental. Computational complexity theory ultimately isn't concerned about any one particular problem and how to solve it quickly for practical applications—the goal is to understand what is and isn't possible with computation overall and with what resources (time, space). Why solve one problem when you can solve all of them?
But to do that requires really understanding the mathematical structures behind computation itself. If you're a formalist, instead of thinking of infinity as "the limit of large n", you think of it as a concept in a formal system that involves manipulating symbols according to a set of axioms and inference rules. You can use whatever intuitive human-scale analogies you prefer when thinking about large cardinal axioms or the continuum hypothesis, but at the end of the day, all that matters in terms of computability and computational complexity is how exploring the space of proofs derivable from these formal systems leads to a better understanding about the behavior of Turing machines (and thus the nature and fundamental limits of computation).