You’re overthinking this. Any athletic shoe is fine. As is going barefoot, unless you have foot problems.

I have some reebok's HIIT shoes I like

I go barefoot on a pilates mat, which is thicker and less sticky than a yoga mat. The mat provides me enough cushion and support.

Search for cross training shoes. Those will be what you want if you don’t want to go barefoot (personally I don’t, my arches said no)

I also go barefoot.

I do my Peleton strength workouts barefoot on a 5 mm yoga mat. Even the cardio things like burpees or skips and jumps.

I recently bought Mizuno TC-01 shoes for my Peloton strength workouts. I was using my running shoes but the soles ar too thick and therefore not stable for the type of workouts you describe. My friend uses some Reebok CrossFit shoes for the same workouts. The Mizunos are very thin soled and you can really feel the floor underneath.

This is my option at home. It seems like too much support is actually worse for home-carpet workouts, and something more flexible would work better? What indoor-only sneakers do you use for at home cardio workouts, that may or may not include burpees, plank jacks, zumba, lunges, among other things?