I was in the same boat and after some Googling I found that you need to have "Displays have separate Spaces" checked in the Mission Control preferences. For some reason it requires you to log-out and back in after activating it, but afterwards I can indeed use split view as described.

I don't remember changing this setting, but it's always possible I did years ago. I have never been a big user of Spaces and somehow missed out knowing Split View was ever a thing, but it's actually pretty neat.

Here's the relevant Apple Support article

https://support.apple.com/en-us/HT204948

Works for me, long click, it resizes to left half horizontally, then when I release it fills vertically. On the right side is a collection of my other windows, I tap one and it fills the right side.

Alas, it's not resizing the windows, but creating some sort of new virtual desktop and sticking them in it.

That kind of turned me off buy maybe it's worth giving it another go.

This is true