I certainly think you can find horses to both halter AND drive... I have always looked for horses that I could do both with.
Now granted... you may not find a National Grand Champion quality halter horse that ALSO is a National Grand Champion driving horse... but I think so!
The minis have SO many driving divisions these days that if you get a good halter horse that has decent motion, you can find a driving division that it will fit in.
You will just have to be patient and look... and try shopping at farms whose horses excel in both (just off the top of my head, check out Buckeye WCF or Taylor Pony Farm... their horses, both ASPC/AMHR and AMHA/AMHR or AMHR only are horses that are conformationally balanced AND usually are bred to move... and the show records of many, many of their horses prove it. I haven't bought a horse from either of those places, but wouldn't hesitate to look. They aren't ridiculously expensive, either.)
There is nothing that a driving horse has, that a halter horse can't have... or the other way around. Just sometimes you can "overlook" things in a driving horse such as a plain head or cowhocks, that you can't in a halter horse. A pretty horse is a pretty horse.
And I disagree with the above statement that a driving horse cannot have a high tail set, and has a rounded tail set...
My stallion has a very, very short croup with a high tail set but a pretty deep hip. However, he can still move. (He is not a mini, but conformation is conformation....) All of the angles together are what dictates what sort of movement your horse will have.
Anyhow...
This AMHR gelding was National Top Ten halter for me, Reserve National Champion in Liberty for me, and then went on to win National Grand Champion Pleasure Driving for his new owner. I wish I had better pictures of him in Halter from Nationals, but I was more poor than I was now
This is his full brother. I loved the above horse so much, he was so versatile, I got another clone
He was National Futurity Champion in halter, I broke him to drive, and then he went on to Hall of Fame in Roadster for his new owner.
I had fun with this little guy (my only AMHA registered mini I've had pretty much), he was National Champion in halter a couple of times, and then he was a super cute driving horse.
I showed this guy locally in halter and driving, and went on to win National Top Ten Halter, National Champion Liberty, and multiple National Top Fives in Pleasure Driving:
Anyway, I just wanted to show you that some people DO like to have versatile horses that can show in many things, even at the National level. And go for geldings! They are great! Sometimes you can find been-there-done-that show geldings that are broke to drive and jump, that the owners just aren't showing any more and want their horses to have a job. And they are NOT always expensive... MOST of the horses I've had, I've not spent more than $1500 on. But it takes a good eye and a lot of patience. If you don't have a good eye yet, find someone who does, and develop yours while you go out there and show.
The more patient you are, the better your chances of finding some truly knockout horses will be.
Good luck to you, and be sure to wait for what you want!
Andrea