Yep, I am familiar with this scenario. Though I will say that I waited until my daughter was 11 and had been taking riding lessons for 5 years before buying her a horse. However, I love horses in a different way than she does. And that's ok.
First thing I would say is that I would never, ever, force a child to go to a riding lesson. One of the pieces of advice I was given on this forum was to wait for her to ask. I started going to the barn by myself. Nearly 4 years later, I bought two more horses, built a barn on my property, and still go to the barn by myself. Honestly, I love going alone. It's my getaway. I'm a very solitary person, and it's hard being a mom/wife all the time. I miss my space, so I'm not sad to go by myself.
Ultimately, you have to let her decide. I think that for many kids in this situation (with a horsey mom), interest comes and goes. Part of the reason is that kids need to develop an identity that is separate from their parent. They have to figure out whether they are really into something, or whether they got into it because of outside encouragement. The only way to find out is to take breaks from it. Allow that and be ok with it. Like REALLY ok with it, because you say you are, but it doesn't really sound like it. (sorry, you said you wanted us to give it to you straight) For the first 3 years or so, my daughter would only take lessons sporadically. She would do lessons for about three months, then stop for whatever reason. She'd find a new barn she wanted to try, and did lessons for the summer, but wouldn't want to go in the winter. My point is that this is totally normal for a kid. Eventually, she got to a point where she wanted to ride year-round, compete, and this year, she was one of 6 riders in our province to receive a 500$ bursary to go towards lessons, so now she's even paying for part of it herself!
I would tell your daughter that maybe she could take a break from riding. Don't say it like it's a punishment or like you'll be mad. Give her another option like maybe she can have a playdate while you go to the barn. Give her a choice, and if she chooses the playdate or other activity more often than riding, stop asking her if she wants to ride. Let her ask to go. If she doesn't, don't bring it up. In a few months, consider leasing out her pony if she has still not shown any interest in it. She may still come back to it, but if she doesn't, consider sitting down with her and telling her that it's time to sell the pony.
Parenting is so hard, and sharing a passion with your child, while it seems like a great way to bond, can also be a huge source of tension. If she does come back to riding and horses, see if you can develop separate interests within that. For example, my daughter became a show jumper whereas I have never jumped - at least not intentionally! I love doing liberty and ground work, I have an interest in nutrition, training, and just everything about horse health, but she just wants to hop on and ride. I do most of the barn chores, but if I have to go away, she's in charge. And now, she has started to give me riding "lessons" because I haven't been able to keep up with mine since her lessons are so expensive. As a result, she has far exceeded my riding level and is now working on becoming a certified coach so this is great practice for her. And you know what? She gives great lessons! I call her "coach" and try to treat her with respect and forget that she's my daughter during our lessons. Not saying this would work for everyone, but I can tell you that if I had forced her to go to lessons, we would not be here today.
I'm pretty sure my daughter will continue to ride into adulthood, but even now, I always let her know that if she gets tired of it, or wants to stop competing, or just enjoy her horse in the backyard, that's all fine with me. I have told her many times that I'd have horses no matter what, and that her horse will always have a place in my barn. She is 14 now, will get into boys soon, then university, then who knows. But I will always treasure the time we spend together with the horses whether or not she remains as passionate about them as I am. Maybe she'll do lots of other things with her life like I did, and come back to them in her 40s like me. That's ok too.