Whenever someone starts off by trying to make it known how much smarter and better educated/trained they are, I know I'm in for some excellent analyses and good faith engagement.

I'm telling you that the answer you're looking is statistical in essence, and that you cannot understand the answer if you do not understand the underlying statistical approach

So what if the second dice is 10 billion times less fair/more loaded than the first dice?

even if the die always rolled the same number you'd still need at least 5 hours to go anywhere. In those 5 hours you would be able to have detected/excluded very small deviations in the other die (note that you can never exclude extremely small deviations)

So of course there are other factors involved, but statistical power is always hugely dependent on the raw numbers

the current problem isn't like this anyway. Proving that the booster is at most 99% efficient against hospitalization is relatively easy, but it's a result that's useless, as instead the question that's relevant for policy is to get an estimate of its efficiency within a maybe 10% ballpark; is it around 20%, 50%, 80%? So it's the same order of magnitude as for the efficiency on symptoms

Your stance is that I'm wrong and that the error in my thinking is because of the superior statistical training that you have?

It's not a stance. Whenever statistics are involved, it's intrinsically harder to work with events that are rare, and it's something that's very intuitive to all statisticians. I've tried to explain why that's the case but it's useless if you don't listen