I don't think this is 100% black or white question, so some food for thought below. Sorry this is a bit long...
12 riders in lessons is not great, instructor won't have much time to look at each child, but there are instructors who can handle this better than some others with only 3 kids...
The costs of running riding schools are incredibly high, and increasing month by month. This means that a lot face the choice of overcrowding the lessons / closing down, so not surprised this is happening.
There are circumstances where large lessons might be ok as well IMO. Lead rein lessons with all riders having experienced leaders (eg. working students).
Also, if your kid is confidently & competently trotting independently after having ridden for two months the centre must be doing something right, this is pretty quick progress!
20 min to get on horses - is this instructor just quickly legging them up or 20min of them encouraging the kids to do their own girth / stirrups? If first this is quite a chunk of the lesson time. If the second, this is invaluable skill to have and 20min for group of 12 is pretty damn quick!
Spending 5min in circles - this is normal, horses need to warm up, and riding circles in walk teaches controlling the direction their pony is traveling, which is a skill they need to learn. Your child will spend a lot of time riding circles in walk at any reputable riding school. If you / he just wants to go fast, I suggest taking up a different sport...
To be honest, 30min of actual riding time is not awfully bad at this level. Think of the stated lesson time as 'how long can I have staff to assist me for': Leading pony to arena, mounting, warm up, lesson content, cool down, dismount, lead back to the stable. A lot of places who don't count leading pony to arena run lessons to 45min rather than an hour. (Assuming yours is an hour from my maths).
Restricting your son's process, I would have chat with your instructor about this. What your 8 year old kid (who has been riding for two months) thinks of his own abilities / need to progress might not match with reality and what would be safe. You said you are non horsey, so I am assuming you would not be able to judge 100% either?
However, having kids younger than 6 in group lessons and only getting about 2-3 min trotting is not great. If this is recent development, I would run this past the owner and ask if this is just a temporary thing or how could it be fixed. Where as it is perfectly fine for lead reins to trot to the rear of the ride one by one, the confident trotters not requiring hands on assistance should be trotting as a ride.
I'm really surprised any riding school insurance allows 2.5 year olds in group lessons like this!
When your son wants to compete, you are likely to have to buy your own pony. Unless you just mean small in-school competitions?
Pony Club, if they are 'real' Pony Club Centre it might be that kids need to have their own ponies to join. If it is Pony Club linked Riding Centre, and they can ride school ponies, it might be that they have max. number of members they can accommodate each semester. Or minimum competency level.
Easiest thing to find out would be to ask the person who runs the Pony Club at the riding school.
Hope this helps!