I think if you can financially afford a horse, have a safe qualified place to keep it and are fully aware of the responisility of owning a horse why wouldn't you? I agree that having a trainer involved, from buying to continued lessons for both you and your daughter is very important to getting the right horse for both you and your daughter.
That being said she is a four year old and things might change so I would recommend a horse that you will enjoy just as much so if she goes off to a new love then you can still enjoy, also having her do "chores" and caring for her horse is extremely important so she is learning responisility.
We can all say what we want but we do not know you or your full situation, I highly recommend having a serious conversation with your husband, daughter and a trusted trainer regarding if it's really a valid appropriate option.
Best of luck