But I will guarantee that you can't just do a simple multiplication to turn a T distribution into a Z distribution.

One could multiply the critical t value by some constant to turn it into the corresponding critical Z value for a Z test and then we could pretend we are doing the Z test - which seems to be what you're trying to do but your approach doesn't actually work. But really what is the point of that? It actually creates more work. And if your goal is to get to that point where you're talking about type-II error rates it does absolutely nothing for you because you haven't actually modified the test statistic in any meaningful way such that you change the underlying distribution.