If you are truly willing to leave anyway, then there's actually fairly little to be lost by having a frank, closed door discussion in which you say that you need some personal time, first as a general request and if necessary with increasing insistence and detail until you are understood. While this would be on on the border of ordinary norms of continued employment, you are in an executive role, in a less traditional company, and you aren't really asking for continuation anyway.
Worst case, you get fired on the spot. But then you were the one who wanted to leave, and your boss is the one who wanted you not to, so that's something only likely to happen immediately as an emotional outburst. Your request could of course cut short your future opportunities with the company, but then you've already done that by previously indicating a desire to leave.
Middle outcome is that you take some time, come back in January, and things still aren't working better, so you plan an orderly departure together.
I don't know what his options are.
Ideally, your boss realizes that (as it sounds like you are in the US) while it's generally important to set an example of trying to keep things running normally, at some level there's a reality that with two successive mid-week holidays, absent emergencies relatively limited routine work is likely to get accomplished over the next three weeks anyway. That could lead to a simple economic conclusion that there's an advantage to giving you some time in the hope that you'll be able to make a greater contribution in the new year, and that the major practical challenge will be explaining your absence in a way that doesn't open the door for everyone else - but then senior roles are normally out of the office for many reasons.
You could also include the possibility of being on call in case of a genuine emergency, if the urgent energy of such a situation might bypass your current difficulties. Or if you could still have some routine involvement such as running standups via phone that could be an option to present, too (even if you can't actually arrange it so that you'll be making that call from a lounge chair on a beach somewhere warm)
And whatever you do, try to get in some sort of break this weekend - not necessarily holiday related, just different. Go for a walk, read a book, see a movie, hang out with friends, visit relatives... try to see if you can step outside of worklife concerns for a moment.
Others have mentioned seeking professional counseling; that is certainly an option as well, fairly orthogonal to what is covered here in that you can pursue either independently of if you are also pursuing the other. You're likely in a better position to address health issues while still nominally employed, but the role you have described hints you may also have some flexibility to survive a gap which others may not enjoy.