Does the Halting Problem have practical relevance? I can calculate all outputs for a finite number of states and inputs
Indeed, if states and inputs are finite (and the tape is finite, which is quite reasonable), then you can easily reduce the Turing machine to a Finite State Machine. However, this is completely impractical, since it doesn't yield any reasonably effective algorithms. I.e. your solution for the halting problem will be to explode the states of the TM with all possible input combinations of size N, and then find a path that reaches a stop state, right? Probably don't want to implement such a check in your compiler.