It's hard to say. DD2 was 6 when we bought dpony and had been at riding school since she was 4.She was cantering and jumping by the time she left,so just carried on the same at home.They never put them on the lead rein where she had lessons,so she hacked out and did everything without being led.When she was 7 she went to pony club,where 6-7 year olds were cantering,jumping and doing cross country-but these are children who were riding a lot at home so bound to be more confident.

Has your DD said why she doesn't want to canter? It might be that once her friends move up,it will give her the nudge to try.Or she might be happy to carry on as she is for a while. Have you asked her if she'd like a few extra lessons?

Don't pressure your DD. Mine was the same. She had her own pony from 5, but one she inherited rather than was picked especially for her. The pony wasn't the best and she struggled with confidence in her riding. We just let her go at her own speed until she became confident and able to deal with her pony's foibles.
Miles in the saddle are far more use than meeting milestones with everyone else. Pressure can be very counter productive and ruin their confidence. If she is happy to potter along doing things at her own speed then let her. DD is a great rider now at 15. She can sit on most things and cope. She's happy with napping and horses being stupid, will try most things and is very calm.
I'm on my phone so can't check back as to whether your DD has her own pony or uses a school, sorry, baby brain but also remember, schools are paid to get
your child doing as much as possible in as short a time as possible. It looks impressive and as if they are earning your money. It's not always best for your child.
Go with the flow and don't push.

What saggy said

She needs to be fully confident

When dd1 was 7 she was very anxious, would only walk and trot - at 15 she hunts and events

Dd3 is 7 and hacks her sisters large, fizzy pony out off the lead rein, cantering off and jumping logs and has won a few show jumping rosettes

All kids are different and confidence is the key

I wouldn't rush her. DD started riding at 2 but didn't start cantering successfully until she was nearly 7. It was only when we got her a pony (a Section B) who was a first ridden, and therefore used to cantering, that she improved herself. Previous ponies were lead reins and she'd struggled. Now she's 10 and has ridden all over the country including at HOYS. She'll get there in her own time with good ponies and tuition.

My son was like that to, the right pony and she move at her own pace.

Absolutely let her do it in her own time - there is nothing more likely to turn her off riding than being forced to do something and being frightened by it. Much better that she's happy where she is and allowed to move on when she wants to.

I'd be tempted to move to a different school. Different ponies each week is one thing, but you don't get to deal with one thing at a time,which if you arent overly confident doesn't help.

From my experience in riding schools, 6yo children have very different levels of ability. At that age, many still struggle with the co-ordination to deliver aids successful, and the strength needed to manage a pony. As she gets older, I imagine she will suddenly greatly improve. I don't think there is any benefit to forcing/pushing her to canter before she is ready. One day she will ask to do it.

However, a private lesson on the lunge might help? Some schools will even have assistants fit enough to run alongside the ponies in canter on the lead rein which can really help with confidence. However, you do need some fit and willing people to do this- not everyone can manage it.

I'm not sure riding different ponies every week is ideal. Riding the same one all the time can cause problems when it's time to move up, but it's also nice to have a few lessons to build up a bond/trust with the pony. Have you asked the YO/RI why they have this policy? I'm sure they have a good reason for it, and you can't dictate what they do with their own horses. If she's otherwise happy at the yard, I would stay there.

If she is otherwise having fun, I wouldn't worry too much. It will eventually come, or it won't, but children's confidence is a very fragile thing, and once damaged it may never fully come back. At 6, I would just be focusing on her having fun!

There is benefit in learning to deal with lots of different ponies and some young riders can become reliant on a single pony and not want to ride anything else. There can also be competition/jealousy over who gets to ride which pony sometimes, so I can see why this policy might make sense for most children.

However, I think there is no harm in asking if your DD can ride one pony for a few weeks (maybe in a few private lessons if needs be) to see if this will give her the confidence to canter.

Have you considered lunge lessons or similar? Maybe she needs a little longer in the school to build up her confidence with one pony? I think 30-40 minutes is normally long enough for small children but maybe riding for a bit longer sometimes might help her. I am guessing they only do canter in the school, otherwise she would get left behind on the trail? 20 minutes might be a bit short for her to build herself up to canter?

I'm not saying this to suggest you really push her to canter, but I think they are ideas worth trying if you think it might help. Have you asked her why she doesn't want to canter?

Anyway, I am sure it will come when she is older, and she may not mind being left behind if she knows moving up means doing more fast work/harder exercises.