Space

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?"

A couple notes:

First, renormalizarion was hand-wavy when it was first introduced, but it has since been made mathematically rigorous. Additionally, renormalization is a mathematical process to make a theory self-consistent. If you consider it an odd idea because it is physically nonsense, I would caution against forming a physical intuition from any given accurate mathematical model. Especially with fundamental quantum mechanics—there’s a reason why there are several interpretations of QM and have been for a century.

Second, and arguably more importantly: this ScienceDaily article is extremely misleading. The original paper (linked by OP in another comment) says

This is a scenario where the inflaton does not exist, and thus opens up the possibility to provide a picture of inflation that is model independent

So the paper does rid itself of the inflaton field, which is, as you said, a bit of a hand-wave. Crucially, however, it does not abandon inflation—in fact, it explains those “for no reason”s that you mentioned.

i don't understand how super fast expansion following the big bang can be described as "for no reason"