Lubrication isn't a huge issue, bikes only have a tiny drip fed chain oiler, exposing them to the elements (most bikes aren't ridden through winter) I guess will shorten service life.
Having sprockets made is costly and you'd need a position to mount a tensioner on (probably engine mount), also have a sprocket made for the alternator. Chains, unlike v-belts- don't tolerate bad alignment at all- so will snap (with potential for alot of damage) if everything is out of true.
It's not a massively stupid idea, but it's not really required, chains have good strength for their cross section, so are required in some applications (i.e Jap Superbike vs Harley Belt driven) when space is at a premium. But when you've already got a toothed belt, ribbed belt, and v-belt option, I'd not bother for the cost vs benefit.
Also with minimal load on the charger (at idle) you'd probably get a fair amount of 'slap' which is noisy. Probably way quieter than German toothies!