No kids here, but my instructor has a son (I think he's about 4), and he does basic lessons on a pony - he may still be on a leadline.
I was crazy about horses when I was about 6 - really wanted to ride. Never got in contact with my current instructor, so I never started lessons. I still wanted to, though. Always was a dream of mine.
Fast forward to teenager-dom. Decided I really wanted to try it - you only live once! Started in June of last year and happy as a clam now.
I know at my barn, kids (even the littles) are taught everything. Grooming, tacking, caring for the tack and barn, etc. Because of this, we don't get many kids under 8. They usually lack either the coordination, dedication, or maturity.
I would take the personalities into account. Will they get bored learning the ins and outs? Will they startle easily? How will they feel if they reach a setback?
I'm very glad I waited to ride. My horse is stubborn and crabby - a good horse, but requires a firm hand. I wouldn't have been able to handle him (or any stubborn horse) at 6. I don't think I would've understood how to make myself clear, how to reprimand him. I also would've broken down in tears in an accident - I got thrown for the first time yesterday and immediately knew that it would've been my last ride if I was younger.
I think really appreciating the horses and the barn can come at any age - just depends on the person. 🙂