... But in the OP's case, the self-timer is merely slow, not non-functioning.
It's slow today. Tomorrow it can freeze up entirely, and it is well on its way to doing just that. The shutter is going to have to be cleaned anyway. If the self timer is slow, it is a cinch that the shutter speed escapement is also gummed up. I'd bet $5 that the 1 second shutter speed is a lot closer to 2 seconds than 1 second. Since the shutter is going to need cleaning anyway, doesn't it make sense to clean something that can lock it solid at the same time -- and not to use the self timer until then?
If the rest of the camera works fine, and the only issue is the self-timer, then the camera needs a CLA. If the self-timer freezes the shutter, get it fixed. It's cheaper than another camera if that's all that's wrong with it.
1. The way you find out the self timer on a Canonet doesn't work is: you use it and it locks up your camera. This can be pretty dissappointing when it happens before you ever get to take a photo.
2. Most of the people here repair their own. It can get pretty darned expensive collecting rangefinders if you have to buy a bunch of them
and then have to send them out to be repaired.
3. By the nature of the question, and the commonness of the camera it is pretty evident that the OP is not an expert camera repairman.
4. The Canonet would have to be one of the very worst possible choices for an amateur to start learning camera repair with (the Hi-Matics and Moskva/Super Ikontas are just about the only ones that occur to me that might be a little tougher).
It doesn't make sense to me to avoid using the self-timer on an old camera purely out of superstition. If it's broke, don't use it or get it fixed. If it works, use it.
Superstition? Remember that we are talking about someone who has likely never worked on a Canonet here (beyond maybe flushing the blades). I propose this as a general rule of thumb: If there is something you won't really need to use, on a camera that has not been cleaned, and if you don't know if using it will turn your camera into a paperweight, then until you do get the camera cleaned -- don't use it!