I would foremost find an instructor who is experienced with young children. You could have the best and the brightest of instructors in the world, but if they don't have the patience for the short attention span of children, your child will not enjoy the lesson.
Instructors, to me, who are serious about taking you on as a client, ought to give you a tour of the barn and all (if not, most) of its facilities, as well as introduce you to some of the horses, particularly those your child will most likely be riding and interacting with. And try to get a feel for the atmosphere. I love a family atmosphere, you won't always get, but it is worth waiting for to find.
To be honest, the barn I fell in love with back home, I found online. The instructor and barn owner put a lot of effort into a well put together website, and it paid off for them and me. I won't set foot in another barn for lessons or boarding.
As for "backyard barns"? Depends on what the term defines. My barn is no super shiny hunter barn with concrete halls and all the bells and whistles. It's over 100 years old, built from cedar wood, with dirt floors and a metal roof, and has been expanded as well as updated over time. It houses some 40 fat and happy horses ranging from unknown bred rescues to high bred horses imported from Spain and Germany. It's one of the most charming places I've ever been too and is my home away from home.
Think of it not as a failure but as a success in how not to do it.
Don't look in a horses mouth for a gift.