@CaptainThe95thRifles
I also agree with the more recent comments - I'd actively strongly advise against a fit competition horse for a teenager who is used to a plodding hack. A well schooled, mannerly horse - and that would include many cobs as well as more "quality" types - should be perfectly capable of doing both jobs.
However, before making any dramatic changes - given the low level of work your current cob sounds to be in, are you absolutely sure that he isn't capable of some competitive riding? There aren't many horses out there which are truly sound, schooled and fit that can't do low level competition / PC. Would it be worth having some additional input into the current horse?
I thought this too initially but then I read the OP again and wondered if it was more the case that the cob isn't really confident/keen on the more competitive activities being more ploddy/lazy by nature, rather than that he isn't capable per se?
If it's jumping the DD wants to do, it is easier to learn (once you get above the baby heights) if you have a horse that's just that bit scopier to get you out of trouble if you get a stride/line a bit wrong, plus it's a lot more confidence building (and importantly, fun!) if your horse is actively enjoying themselves and taking you to the fence, rather than you having to push them all the way.
My little 14hh cob mares enjoy their jumping and if correctly ridden will get round an 80-90cm course clear but they are quite 'unforgiving', you have to get them just right at every fence and hand-hold a bit, they're at the upper limit of their scope at that height really, if you're a bit long or just faint hearted they will go 'eep' and stop or clatter their way through for multiple poles. It doesn't bother me as I'm not a jumper really and don't care if we rack up a cricket score, but I can see for a teen just learning the ropes themselves and without the experience to do that hand-holding they could easily get a bit disheartened and into a confidence loss spiral, esp if their PC buddies are flying around clear each time. I think similar could apply to dressage at ele or above, where again with correct schooling, it's possible but you would be pushing against a cob's natural build and inclinations to try and get the higher level movements?
I suppose it's an attitude thing, I can totally see that learning on a horse where you really have to ride correctly (as opposed to something more push-button where you can sit there and look pretty) is really good for your riding in a way, but I can also just see the argument that if the cob in his heart of hearts just wants to plod around hacking is it really helping anyone to try and turn him into something he isn't?