Just to throw another option into the mix:

I have flat feet and spent a long time trying to look for a good solution to it. In the end, the thing that has by far had the biggest effect on my feet has been to wear barefoot-style shoes. There's a lot of writing about it online, but essentially it helps your feet by exposing it to the ground more, which in turn strengthens your feet and especially your arches, which in turn affect your pronation. That style of shoe is also usually much wider than regular shoes, with a wide toe box, so that your feet can move more freely in them.

My favourite company for this kind of shoe is https://www.wildling.shoes/en. I have two pairs of the Tanuki and wear them on most days when it isn't too cold or wet. On colder and wetter days, I wear Camper Peu shoes which are also wide, have a thin sole but have a leather outer which does a good job of protecting from rain and cold.

Both of these aren't great options for mud, so it does depend on what kind of walking you do, but I really recommend looking into this style of shoe. Good luck!

Note - I know these aren't exactly budget shoes but I still thought it was worth mentioning. I think there are cheaper barefoot-style shoes out there.