It would also be good to mention that as of now, the COE website does not offer visa exemption/bilateral agreement as an option. The embassy jumped the gun.
Even with all the troubleshooting, it's also possible the online system just won't work for you for some reason. It happened to me (same error message) and I had to go in person to Muang Thong Thani to file my 90 days.
It's called an Embassy letter because it's issued by an Embassy, not a consulate. Just like in any other country, the US has only one Embassy in Thailand, so it's perfectly normal.
For what it's worth, a half page would be enough for entry/exit stamps in Vietnam, Malaysia, or Myanmar (e-visa only). But you'll have to hope the immigration officer is not a stickler.
Obviously, getting a new passport would be much better, if you are able to.
I've been on non-O volunteer visas for 3 years. It's a straightforward affair as long as the paperwork is done right.
There is however a peculiar process regarding extensions. Volunteer visas are usually extended for 90 days only, because 1-year extensions require an agreement between your foundation and an official government agency. It's rare and difficult to secure.
Another thing is that immigration officers are careful about people who could misuse this visa. I regularly have to answer a few questions upon entering the country, just to make sure I'm not a freeloader.