I'm going to go in the other direction here, as I usually do on this particular topic. I say if your daughter is showing some level of talent and love for the sport and JO level team gymnastics is the goal, then I say start her out with the long term in mind. I'm correct in remembering you saying that this gym does not have optional gymnasts right? If that is the case then how quickly the kids move through the compulsory levels is of no concern to them, but that doesn't mean that is what is best for your daughter.
Now, that doesn't mean that you need to run out to the closest super elite 30 hour a week for 8 year olds gym, but a gym with a solid program, good foundation work and the opportunity for high level gymnastics (meaning 8,9,10) I would say is a must if this is the road your daughter wants to go down. Just realize as a parent what a commitment to that path means, not only for your daughter but the rest of your family as well. If you aren't ready to make it then stay where you are.
As for her age, yes she is young and there is certainly no reason to rush the process, but she is not still in preschool and at most gyms I know of at six she should be moving toward competing new level 3/4 in the next year or two.
It really comes down to what your daughter wants from gymnastics and what options you have available to you. You need to find the right fit for you kid right now, but with an eye on where that might lead. Just my two cents.
Good luck with your decision