LiIon batteries can be damaged if they're actually completely discharged, but the ones in ds/3ds (and any modern battery really) have circuitry to prevent this - if the battery can't recover from being discharged to the point of no longer providing power then either the circuitry or the cells are already damaged far beyond the point of no return. Though, granted, discharging to that point isn't great for them, so you should avoid doing it regularly.

Charging to full isn't always enough to completely recalibrate the battery. Battery management circuits typically keep track of both maximum and minimum charges, and both can get out of whack, so only a full charge may or may not solve the problem.

I double checked all that with google, I'm on mobile so linking a bunch of sites is a pain but you should be able to verify with a few searches (if not lmk and I'll find the links later). Anecdotally I've personally had battery issues with an o3ds (and a couple other non-nintendo devices) that were solved with a full discharge-recharge cycle, but not with just charging to full, so regardless of the reasons it's undoubtedly helpful in some cases.