The OP is asking about when to introduce lateral work.
That has not much to do with the question. Kids growing organically with animals would probably not even know about « latetal work » … the horse might go sideways but not in any fashion way whereas the kids who would take lessons and leave actually would/could.
Back 50 years ago, people might have had a backyard horse. Kids were grew up in what is now termed “free range”…galloping on the back of a pony…without a helmet. That life has become much harder to impossible to find now.
Doesn’t mean their riding was any kind of good… again, I fail to see how this is related to the OP, except from the fact that the OP is actually doing exactly what you are saying has disapeared, having a horse on it’s own, yet not knowing what to do with it… which kinda backs up my claim - the need of a trainer.