It just depends on how remote an area I'm riding and how long I expect to be gone.
I leave the cantle bags on my saddle all the time. In those, I have a waterproof container with matches, a space blanket, a very small first aid kit with asprin, chapstick, a few bandaids etc, and there are usually one or two granola bars, small bottle of gasoline to start fires with. I almost always carry two water bottles.
On a ride by ride basis, I will add to this as I think I may want. If I'm going to be gone all day, I'll slip in an apple, string cheese, a sandwich, more granola bars etc. Depending on the elevation and weather forecast, a jacket or raincoat. It's no fun to be 3 hours from the truck/trailer and get caught in a rain storm. At least with a rain slicker I can stay mostly dry.
In my career of riding horses, We've had to put one horse down during a ride. So I usually pack a pistol on any kind of remote ride where help would take a long time to get to us.
In my trailer I keep many of the basics. A Rasp to trim hooves with, chaps and chinks for cold or wet weather, several jackets, Hats and helmets. I keep several of those LED flashlights. If I think I may be out late, I throw one of those in. I always have sleeping bags and blankets in the Gooseneck, in case I have to spend the night. I also keep a bag of horse pellets. Again in case I have to spend the night some place, I can feed the horses. I also keep some extra horse gear. A high line, extra leads, hobbles, extra reins, cinch etc in the trailer. We have a bigger First Aid kit in the trailer. A friend of mine is a Surg Tech, so he put together a really complete kit, We could stitch up a horse or person if we had to.
I ride some extremely rough and remote places, you may not need as much stuff for a ride down the trail behind the barn. But where we go, there is often no cell service. It's often wilderness and no access with a truck or other emergency equipment. If we break a leg or arm, we still have to ride the horse home.
And no I don't use any leg protection for the horses. A good trail horse should know how to protect his legs.
You may want to attend a NATRC competitive trail ride. Even if you just go up and volunteer to help for a day. I think you would learn a lot about trail riding, by watching what the judges look at. Check out their website at natrc.org and see if they have any rides close to you.