I ride in Asics. Running shoe. No heel. Don’t care if it has a heel, or not.
I have found no other comfortable riding shoes. Also these are great, get off and walk or run, comfy ALL THE TIME.
Worried about no heel on footwear, get cages like BHLH suggested.
I have plastic black stirrups with lovely cushiony foot beds. The more cushion the better.
They do not make “boots” like they used to, so I wear running shoes only these days. My ariat terrain half chaps fit just perfectly with the running shoes.
And I can wear my running / riding shoes any where besides just on a horse. I have muck boot shoes to clean stalls, and other shoes to do other chores.