With the huge number of stories of people getting themselves hurt or killed taking selfies with their cell phone, you would think that a self timer is mandatory. Yes, they often jam if they haven't been used in a very, very long time. But, guess what? Such a camera needs CLA overall.
Virtually every Leica camera that I've bought (and I only buy used) has required an overhaul. So, plan on the expense of having a service done on any camera that you acquire, unless there is documented recent servicing. Even my M7 I sent to Leica NJ to have the optical film reader and the flare reduction optics placed in, along with a general CLA.
Parts availability for the M2 and M3 are starting to be a problem. I bought an M2 (has ST and lever rewind) which had been a demo camera for a German camera store. It had stayed in the back room self for decades and was finally sold when the store closed down. It looked cosmetically in EX+ condition at least, but functionally it was a basket case due to disuse and congealing of the lubricants. So, I sent it to Gus Lazzari. This was along with a truly mint M3 in box with L seal, which I obtained at ridiculous prices from a Tamarkin Camera Auction, and a mint minus M5.
Two years later, he finally finished servicing all of these cameras (he has a seriously long waiting list, as does Don Goldberg). Prior to sending the cameras back to me, Gus mentioned that the M2's self timer pan screw had been marred by someone using the wrong tool to remove it in the past. So, I searched high and low for such a screw. The one available on ebay had corrosion on it. It must have been taken from a camera that fell into salt water in Japan.
Finally I called Don Goldberg. After begging and pleading on my knees, he relented to selling me an original Leica M2/M3 self timer pan screw, new old stock. Even he is running out of these things, and Leica no longer makes them. So, had I not been a good customer of his in the past, I doubt that he would have sold it to me. Don also gave me the admonitions:
"It's a left handed thread. Also, if you don't have the correct tool to mate with the two divots in the screw head, you WILL mar it". I had Don send it directly to Gus.
The cost of servicing these cameras and an MR-4 light meter (composite of two: one beautiful case with battery acid corroded circuitry, and another with an ugly case but intact circuit board) would have allowed me to buy a very nice condition M6.
So, the moral of the story is to mentally budget $500 or so for overhauling these cameras. Then be prepared to wait. And, sometimes the M3 and M2 no longer have parts available.
Hence, my original suggestion to go chase after an M4-2 or M4-P unless you really have some pressing need to have an M2.