There's precisely one weapon in the third game that automatically recharges -- a very late game rocket launcher equivalent, which even if you equip your entire team to the max with (which would be something like 2-3 of them per squaddie) isn't enough to finish a mission at that point. They hit like a truck, but you only get something like three shots with them, they're terribly heavy, they take up a huge chunk of inventory space, and they're pretty slow.
Plus, iirc, you can't actually build them. They can come on shipments from earth, but can't be built locally. They're pretty useful, but not something you can mainline as a weapon. Strictly support.
... anyway, the definite thing I like about the third is, among other things, that a pistoleer can be basically the deadliest thing on your team. It's pretty glorious to hit the end game and have your highest performing squaddie be the one that's been going gun-kata all over the aliens' face.
Though yeah, ammo isn't really an issue beyond the very, very early. Once you've got decently trained engineers and some better suits, bringing along enough ammo to murder everything that gets generated on a particular mission is cake.