Hmmmmm.......
So, I'm either Infantry or cavalry.
Think I'd rather be cavalry. I've been Infantry. They walk too much.
Interesting so many like a shotgun.
I'd feel severely undergunned with that as my only long gun. What if I'm facing a good rifleman under cover or concealment? How many 12 ga slugs are you gonna hump? A rifle will cover shotgun ranges. A shotgun will not cover rifle ranges.
Given a choice of engaging at 30 yards or 150, I'll take 150. He might not be as good as me. If he's as good as me and saw me first, he'll probably own my gear and I won't be worrying about it.

I'd rather have a 22 rifle rather than a shotgun if those were my only choices. I grew up a rifleman.
So.........
If mounted, a 308. A bolt or an FAL would work. A 30-30 would work. Any HV centerfire down to 223 would work, but the most common calibers would be better, so I'd stick to 223, 243, 30-30, or 308. If the horse blows and I have an FAL or AR, I take two mags and loose ammo. If you walk into something two mags won't handle in the short term, save the 40th round for yourself.

And yes, I only do 20 rd mags.
No 30's.
Too long.
Keep me too far off the ground.
Speaking from experience, belt buckles and trouser buttons are too thick, and keep me too far off the ground.
If walking, an AR-15 or 30-30 or light bolt gun at 308 or less and a 4'' K frame 357. A 9mm would work. My Sig 220(45) would work.