There's nothing handwavy about renormalization, it's just a way of describing the mathematics which is easier for a human brain to deal with, so we've standardised on it.
An unnormalised wave function can show you the relative probability of any given thing, but it makes like easier if you set the scale so that you can read an actual probably straight off it, rather than having to ask "relative to what?"
It's only true of badly designed bridges, these days. Modern engineering tools can calculate the resonant frequencies, and they make certain that those are far away from the frequencies which humans or wind can create