A good way to check for poorly documented features is to search reviews. I didn’t find any that mentioned a power button/switch on the MS 2000.
Tangentially related anecdote but I’ve been using a Logitech K400 wireless keyboard/touchpad since Feb 2017 and it’s still on the original batteries. Granted, I use it very infrequently (it’s on a laptop that’s one source for my television) and weeks go by where I don’t touch it and when I do, it’s brief periods of trackpad use. It has a mechanical sliding power switch that I never use.