Ah yes, the old battery versus mechanical argument, my old foe...
Yes, the mechanical cameras will work w/o a battery. Thank the gods for that! But unless you live in a battery-less black hole somewhere, that's not really an issue is it? Is it really all that hard to carry a spare battery? And do you think mechanical cameras magically stay in-calibration through the years? Any OM-1 you buy for less than a hundred dollars will need a CLA and that alone is over a hundred bucks.
While I admire and use many battery-independent cameras, the electronic ones are more accurate for a longer period of time and the batteries are cheap and available. Go with the AE-1. It's a superb beastie.
Yes, it's the 21st century my Canon A1 has only had to have the battery replaced
three times in the twenty three years I have owned it., and I always carry a spare