I took my mare hunting a few years ago and she did the same thing. I made the mistake of taking her out in the snaffle we rode at home in. My normally super soft, supple, responsive horse turned into a running, pulling maniac. She was definitely having a good time and my efforts to rate her was getting in her way! lol She figured out that she could ROOT down and shake her head in an attempt to yank the reins out of my hands. My back and shoulders were SO sore for days afterwards from trying to sit up and keep her at a reasonable pace. There was no such thing as a half halt that day! lol
On thing that’ll help is bridging your reins so they can’t rip them out of your hands as easy or take their head down any further once the “bridge” hits their neck.
I also had to resort to a kimberwick. It honestly was the ONLY thing that worked. I did my best to school her a ton to try and fix the problem but it honestly went out the window when she got excited. And didn’t seem to tire either, she was as prancy and excited at the end as she was at the beginning, so sometimes more exercise isn’t necessarily the solution. The bit seemed to fix her little red wagon, she tried pulling a few times and then realized that the result wasn’t the same as with the other bit. It only took a couple of times for her to do that before she started listening, after that we had a fantastic time, no more rooting, much more control. She seemed to enjoy that I wasn’t bugging her non-stop and I was enjoying just riding.
Anyways, all in all, I think it depends on the horse.