Whichever direction you go in, I would not be buying a Connie or a Welsh D with the idea that my 3 yo could share it! They are big powerful ponies usually with quite opinionated personalities (which is what makes them appealing for adults on ponies, but makes them problematic for children.)
I have a saintly Welsh D who does the bare minimum for his rider - he would not scare the parent of a small child but he takes a bit of effort for even a very capable rider to get going because he is quite lazy and stubborn and would quite easily catapult a smaller rider over his ears given half a chance when he sticks his head down for a sneaky mouthful of grass. There is not a cat in hellās chance a little rider could either get him going or stop the snacking. By contrast, the more forward going types would probably traumatise a parent simply because their flamboyant forward movement can take quite a bit of containing, again impossible for a 3 year old and once a Welsh D actually gets going there is no way a person on foot would be able to catch up to help a toddler stop it!
Connies can be sharp and clever, they are fab ponies but when I was looking for a larger pony/small horse for my daughter to step up onto the ones we looked at were generally too much pony even for a 12 yo to manage well. The few quieter and calmer ones I have seen have all been much too small for an adult to comfortably ride. I have friends who event their Connies at BE100 level and know several people who compete their Connies (and Welsh Ds) up to international level in TREC.
By all means look around, but if you want your child to be able to handle and ride a pony well, it will not be the same pony or horse you buy for yourself. I have a 15.2hh Connie cross, who I have owned since my DS was a baby, and who he has known his entire life, and at nearly 14 I think DS is almost strong and capable enough to ride him now, under careful supervision. He has been on the saintly Welsh D for the last 4 years, and before that on a 11.2hh Welsh A.
I see many children overhorsed and put off early on in their lives, especially when a parent buys them āsomething they can grow into/learn together with.ā Itās a sad fact, but in order to really learn to ride, a child needs a pony of the appropriate size and degree of forwardness for their age and size, which will obviously change as they get older and more capable. They also benefit from riding a range of different ponies whenever they can, as not all ponies behave exactly the same and you need to equip a young rider with skills to manage different pony habits if you want them to become a good rider. It is not possible, no matter how much you think you want to offer a pony a home for life, to keep both child and pony happy if you try to buy one pony for a 3 year old and hope this pony will last them for the rest of the ponyās life. What is suitable for a toddler will be boring for a 10 year old, and not refined enough for a 15 year old. What is suitable for a 15 year old will be far too much and in the region of possibly dangerous for a toddler. This is before you consider how frustrating it will be for the pony to have to tolerate a child that is not suited to it any more.
Best thing is to buy a horse or pony for yourself that you could realistically lead the toddler around on for the odd ride, but eventually take rides on at the riding school if there are loan opportunities there. It will be a much better way of getting your toddler riding on something a suitable size without putting you in the position of having to consider selling an outgrown pony if that is how you feel.