Please don’t put lava rock all over that slope.
It would look awful. There are several good stories on Houzz for sloped yards—have you read those for ideas?

Does any of the run off from the hill go into the pool? I think I would create terraces between it looking natural to flat decks where there could be seating.

But water drainage and sprinklers are something else to think about.

I vote some sort of plant material. Rock naturally moves down hill so even if you put it all over it will slowly creep under pressure and you will end up with it at the bottom instead of the top.

We don’t have any drainage that runs into the pool. The water is directed to the other side of the house that works well. I’m trying to avoid digging in that area but maybe more terraces would work. Would some ivy work to help with coverage or something else that could do the same?

If you are looking for a good groundcover to take up a good chunk of the space, depending upon where you are, I really like 'Streib's Findling' Cotoneaster, Cotoneaster dammerii. It is a flat as a pancake, no climbing, evergreen woody shrub with blooms and berries. It must be installed properly (no landscape fabric!!!!!!) and weeded for the first couple of years until you reach full coverage, but long term it is an excellent low maintenance groundcover.

https://www.monrovia.com/plant-catalog/plants/1006/streibs-findling-cotoneaster/