Going under just always has more risks and more recovery time. I just think on a personally level it's not worth it in place of "a mouth full" of needles (it was hardly any) unless it's just not possible to do while awake, and if it's not, it's not and you can't do anything about that. I studied medicine and I just think if one can avoid going under, one should as there's always a risk. That's my 2 cents.
What I'm not sure about now though is is this full on general anaesthesia or just knock out gas? Are you actually going under? If you are, you have no choice in the matter, you can't wear a binder. If it's knock out gas, then you may be able to.
However, even with knock out gas, I urge you to tell the anaesthetist that you have a binder and ask if it is okay to wear it. No medical professional should judge you for this and if they do, you don't want them as an anaesthetist anyway cause they're a prick and never trust a prick with your life! You should be able to privately talk to them too, so you wouldn't need to disclose to the nurses or dental surgeon. As much as you might not want to disclose it, if you were to have a reaction or something went wrong, not being able to properly ventilate you or not immediately understand why you're not ventilating effectively could have way worse consequences. Even though they should check that anyway, it's not worth risking your life in case they take happen to take a few mins to catch on. At least if they knew off the bat and allowed you to wear it, they would be prepared to immediately cut it off if something went wrong.
I'm not saying people in N America are "forced" to have wisdom teeth taken out, it just feels to me here that the whole optional idea is like taking out an appendix in case you do get appendicitis one day. It's quite a risky and invasive procedure for a maybe in my opinion. Here they were REALLY pushing me to get the knock out gas and do it in two stages, and I then started emailing back and forth with my UK dentist, sending him all my X-rays asking if this was really necessary and he assured me it wasn't and put me in touch with a dental surgeon in my home town that told me how he'd be able to remove them in one visit under a local and explained the whole process. 4 different dentists later and I decided to hold off until I went home for Xmas and get the whole thing done there, and I was in and out in less than an hour despite the problems of them being curved under the gum. So while I absolutely did need the teeth removed, that wasn't optional, I didn't need knocked out at all and I was definitely being pressured into doing that. Maybe because they'd make more money that way.