Depends on that cat.

In my experience if you have access points for mice they will get in whether you have a suburban turf grass lawn or not, and a cat can't get them if they are in the walls or crawlspace. So the best bet is to seal up any holes and keep all vegetation, native or not, at least a couple of feet away from the house.

it kinda does both, there are more mice but the more naturalized habitat gives them more places to hide that isnt your house, especially in the spring/summer fall, but winter too. I dont know, others get mice all the time anyway, we occasionally do, i dont know if it's an improvement or not. I do know that a well sealed house in the woods with totally native habitat for acres (not mine sadly, lol) has far fewer pests than in any suburb house so i think there's merit.

I had a cat once that let a mouse come up to it, touch noses, then run away.

It really REALLY depends on the cat.

One of my cats would probably be scared of the mice, and the other would probably make friends with the mice. They are both disappointments to their Great Ancestors