On finitist interpretation of Computer Science

Even though the notion of infinity has been proven useful in a lot of analysis methods today I will argue that sometimes it creates fundamental problems that could have been avoided.
Let's start with one of the pillars of modern Computer Science - Turing's Halting problem. If we do not assume the existence of infinite-tape infinite-state Turing Machines it becomes clear that for all Turing Machines with at most N states and the size of the tape of at most M we can construct a Halting Problem solver.
This solver will construct a graph in which every combination of a state and configuration of the tape is a node. There are at most N*2^M nodes in that graph. Then we can use any suitable graph search algorithm to determine the feasibility of the end state node from the desired start state node.

The next thing to consider is how realistic are those models of computation. We are yet to see a single infinite tape Turing Machine. One can argue that the most powerful computers nowadays can be considered to have infinite memory and thus are representative of an infinite tape TM.