You are correct in this. If I wanted to solve whether a program P halted with input x in some amount of time t, this wouldn’t really be a hard problem to solve. In software engineering this is more about what we care about.
However, in the big picture, what’s interesting about the Halting Problem is that it shows us that…
Awesome! There are a few things I recommend, though this is not an exhaustive list and I encourage you to seek out new sources on your own.
For the computation/math side: