Your quote from the PHB is leading you astray. Those aren’t general rules for carrying capacities a la 3.x, those are rules that tell ‘you’ (i.e. someone statting a PC) how much weight that PC can carry. The design paradigm in 5e is such that they don’t need to make the rules work for sprites or any other monster-like thing, because they know that all PCs are Medium or Small humanoids, and any exceptions to that will be handled exclusively by the race description section, as far as the 5e paradigm is concerned.
Sprites aren’t a playable race, and there’s no subsystem for penalizing monsters for carrying too much or prohibiting them from lifting more than a certain amount, generally. Moreover, since they are Tiny, one should reasonably expect the rules to be somewhat different.
If the DM cares about tracking encumbrance, they will need to come up with rules for the sprite. Probably they will give it a multiplier to how much it can carry and how much its equipment weighs. If they care particularly much and/or it matters, they will reduce equipment weight more than carrying capacity to account for smaller creatures having lower strength scores, generally.
If the DM doesn’t care about encumbrance, they will just give the sprite disadvantage when they are doing stuff that should be affected by carrying too much and carrying what seems to the DM at the time like ‘too much’, and not let it pick up stuff that’s ‘too heavy’, subjectively.