Just some words of advice:
Don’t just desensitize the horses to the stroller, but also an unhappy baby. Nothing set my horse on edge worse than hearing a crying baby even if she was in the tack room or parked in the stroller on the other side of the fence. He wasn’t sure if she was a monster coming to eat him, or a monster already had her and was coming for him next.
I would not advise babywearing and throwing hay. Even feeding, I’d be very, very careful about bending down.
Be aware how hot it gets in the stroller. Keeping the bonnet up for shade is great, but it will hold in a lot of heat. I found a great battery powered clip-on fan with soft foam blades that keeps her cool and fairly entertained.
I try to time our barn visits for nap-time. She’s quite content to sleep in her stroller, as long as she’s comfortable and usually well-fed.
Watch out for insects. I took her to a show and had to cross a lot of grass. I ended up pulling a couple of ticks off the stroller. Thankfully they did not reach her!!! I do spray around the base and wheels with a picaridin spray if we’re going somewhere with unmowed or minimally mowed grass.
I have a Britax, forget which model, that handles everything but footing-deep sand quite well. It handled the cross country course at Great Meadow like a champ, and DD actually slept for all but the last two riders. She was quite content with her little baby-sized neck pillow and fan.