I've seen a lot of 13hh or so ponies jump higher than the big boys. Many of them are very tough and athletic. It's not at all uncommon. Pnies tend to be pretty brave too, so many of them really love the jumps.
The issue is not whether a mature, healthy pony can or should jump, so your coach must be thinking of something other than the pony's size.
At 5, he can go ahead and jump. Is he sound? Any injuries or developmental issues? When he was trained at age 2, was he trained lightly or was there a heavy load or hard impacts that may have caused musculo-skeletal problems? Just trying to figure out what your trainer might be concerned about.
For giggles, go to youtube and type in pony jumping and you'll see that those little guys can really fly!
I learned to jump on Fjords, which are usually in the 13-14hh range. Jumps were mostly around 2 feet, and we maxed out at 3'. Similar build to the Haflinger, I would think? Here's a random pic of a Fjord flying high: https://www.google.com/url?sa=i&rct=...78692310958713
BTW, my 8-year-old daughter takes lessons on a 12hh Welsh pony that's hunter-trained, with lots of competitions and wins under her girth. That little mare is 29 years old, and still teaches all the smaller riders a thing or two about a hunter-course with gusto.