They have touch screens, but it's minimal, pretty much only usable in the GPS view.

They do have a rotary dial and push button that controls the display, so no touching it. Makes it a lot simpler when your UI designers are restricted to left/right movement and button push. Can't hide things all over the place and under deeply nested menus.

The 3 2021 Mazda models I test drove had no touch screens at all, I believe they included an option between the two for a couple years but have now phased out the touch screen.

The new Mazdas have no touchscreen and in my opinion it's worse.

Apple CarPlay and Android Auto are designed to be touched. Navigating either without a touchscreen is a horrible experience and requires significantly more steps than just touching the item you want on the screen.

I found no serious issues using Android Auto with my Mazda, what kind of things are you doing during your drive that are so impossible to do with the rotary controller?

The only thing I can remember might be the address entry, but that can easily be done with the keyboard on the phone itself (click "phone" icon in the on-screen Auto keyboard to handoff).

Funny you say that, because a 2018 Mazda3 is what I currently drive. It has a screen, a small one, and every common cabin control is still a physical button. The only exception is the radio, which is software, but can be controlled with a knob. But I just leave the radio on the same station always so it's not a problem for me.