I see this come up on the AERC Facebook page on a regular basis, so I figured I might as well address it here.
Can a Haflinger do endurance?
Short answer: Yes.
Long answer: Some can. I think most horses could do a LD if properly conditioned, and Haflingers are no exception.
Should I buy a Haflinger specifically for endurance?
Unless you're specifically trying to make a point.. I wouldn't. They're neither bred nor built to excel at the sport.
Don't get me wrong, I love my mare and I'm happy doing endurance on her. But to try to find a Haffie that will both do well and enjoy distance riding at speed, and that you like, and with a good mind? I think there are some that can do it, but it might be really miserable trying to find one that fits all your specifications.
Should I buy a Haflinger to have general fun with and try endurance too?
If you want a Haflinger, there is no reason you can't try endurance with your Haflinger! I do slow trail rides, fast trail rides, dressage, occasional jumping (occasional because of me, not because she's not capable or unintested!), bareback rides, whatever. I wouldn't be opposed to trying her on cows. I know she's had some driving training and some Western training. She may not be brilliant at any of them, but she's happy and capable as an all-around amateur's horse.
Haflingers and speed:
Based on my limited Haflinger experience, I would recommend a GPS of some sort. Figure out how fast your horse's walk is. Figure out how fast their trot(s) are. Some Haffies have big walks; some (like mine) do not, and you'll want to make sure your Big Trot is big enough to make up for that. Personally, Fetti and I trot most of our LDs and only walk when terrain absolutely demands it or when she asks for a break.
A more modern-type Haflinger will probably give you a body type more likely to excel at distance riding.. but there are certainly several drafty Haffies out there doing LDs/endurance, just as there are draft breeds doing endurance!
Miscellaneous notes/things to keep in mind:
It seems that about half the Haflingers are kid-safe and bombproof, and half are feisty explosive firecrackers who may or may not have a screw loose. Confetti definitely errs towards the latter. I joke that sometimes she tries to be an Arabian. Do your homework. Don't assume that all Haflingers are calm, mellow, and sane.
Haflingers are notoriously strong, smart, and stubborn. I've found all of that to be true.
If you're riding a non-Arabian in LDs/endurance, especially one that stands out as obviously not an Arabian? Embrace being different. I practically introduce myself as 'Hi, I'm Fig, and I have a Haflinger'. No one is likely to remember your name, but they'll remember you for your horse.