Agree, agree, agree.
If you want to find something for your daughter to ride, look for a lease of a school pony type. She will outgrow what you need now in ability and size wise in very short order, so the lease of an absolutely bomb proof leadline and walk/trot pony is what you want now, to be traded up for something requires a little more riding in a year or two.
Also beware of the fact that a six year old usually can't handle a horse by herself in terms of catching, grooming, tacking up, etc., and that my interfere with her feeling like it's "her" horse.
With lesson kids coming up in my barn, I wouldn't let them lease a pony until they could catch, groom, tack up on their own, and do some simple schooling in the ring with minimal supervision. Average age for those skills was 10 - 12; very few children are able before 10. I preferred that they lease a pony from me for 6 - 18 months to sort of try on the responsibility of horse care before they bought their own.