I guess I am the only person who thinks that it would be better manners to not ride on that piece of grass. I don't know the legalities of who owns that piece of land, but taking the OPs word for it - even if she doesn't own it, she maintains it, and she has to look at it every day.
Horses do tear up grass just by walking over it. At my barn, which is at my trainers house there is a piece of grass that is a short cut to my horses field. I walk on that but when I am leading my horse we go down the road and use the driveway instead.
If the 11yr old rider isn't safe to be on the road, she probably shouldn't be next to the road either. It would take next to nothing for her horse to have a reaction and end up in the road.
I'll qualify all the following with the fact that I have no idea what a shod horses foot might do. All my comments are based on unshod horses, since we've never put shoes on our's. It could be that a shod hoof does terrible things to the ground. I never paid attention during the few times in my life that I rode someone's shod horse, so I'm not qualified to say what a shod hoof will do. (except that shod feet get worse traction on pavement....that I do know first hand)
I've allowed my mare and filly to graze along the same 1/4 mile stretch of shoulder almost every day since before the start of Spring when I've taken them out for training/conditioning and except for where you can see their tracks on the dirt road at the points they step onto and off of the shoulder (where we start grazing and stop grazing) you can't tell a horse as been on the shoulder unless they happen to crap while there. Even at the 4 mail box areas that the people keep cut and maintained. People have ridden up and down this shoulder for years and I've never seen any sign of it being torn up. There's not even an indicaiton of a "trail" along the shoulder, since the horses don't follow the exact same line every time they walk along the shoulder. Unless you've got a mostly "dirt" shoulder, or the horse is running (which can tear up chunks of sod), or the horse is acting up and moving erratically (which will twist the sod up) there's not going to be much if any indication that a horse was there if they're just waking. Well, unless of course the horse happens to answer the call of nature, but that helps the grass instead of damaging it.
You would have to have an awful LOT of horse traffic on a well sodded road shoulder to start tearing it up. I know from 1971-1980 three horses almost daily going out and coming in never messed up the shoulder. That was from a working farm, so there weren't many days we didn't ride out. Although there were times when all three didn't go, it was extremely rare if none did. (Riding fence lines beats walking them.)
While we didn't mind riding on the road, and usually did ride on the paved road when available to help with hoof wear and hardening, but that piece of highway between the road to the house and the third pasture had a sudden drop where the land natually dropped towards the stream and they rose back up. That made that section of highway blind to any driver until they crested it. Only a crazy person would ride on the highway there. But even after 9 years that I was there with that shoulder being ridden on very regularly it still didn't show any signs of it, unless you caught it on a day the horse left a deposit.
I've seen smaller pastures with turf damage, but that's with a lot of horses being on it for longer periods of time. You get group of horses running or playing around and they can tear up some turf and mess up a small pasture. But that's a different condition than riding along the shoulder.
What it really boils down to is dealing with reality when you select where you want to be. I enjoy things the city has to offer (plays, restaurants, conerts, etc....), but I dislike the "light and noise" polution and I would rather have my horses in my back yard vs boarding them. So the country, where that is normal is a better choice for me and I can visit the city if I feel the need for what offers. If I disliked "farm" animals doing what they do, smelling how they smell or looking how they look then I shouldn't live in the country, but move to a city or the "burbs" and just visit a State or National Park for a pic-nic or stroll, etc... when I want to enjoy the country invironment.
If I have a problem with someone riding legally in front of my house then I should live someplace where it's either not allowed (good luck finding many of those outside a large city), or where it's extremely unlikely to happen. The country is not one of those places.