Yes they did, Fallout 3 is what introduced weightless ammo and a higher starting carry weight.(1 str in Fallout 3 = same carryweight as 5 str in Fallout 1/2)
By comparison the original games has a much more restrictive inventory, ammo weight was the biggest limiting factor, and I'm pretty sure most items were at least a little heavier. If anything New Vegas ammo is way lighter than in the first games. Obviously there's ways to circumvent the weight limit, but it's progression based, same as NV.(In forms of perks, lucky 38 storage, str upgrades)
Just to make things clear:
1. I'm comparing the loot haul potential of FNV with Jsawyer's mod (which basically halves the weight capacity of both the player character and their companions) versus loot haul potential of FO1+2.
2. Loot haul potential = the max weight a party could carry, considering all factors.
3. FNV (unmodded) has 400 max carry weight with all perks and special gear, with 250-300 carry weight within reasonable expectations (a STR score of 5 or so, plus 1-2 carry weight perks). Plus 2x200-ish carry weight for 2 companions. A total reasonable max of 700 pounds.
4. FNV with Jsawyer mod has around 250 max carry weight with all perks and special gear, with maybe 130-150 carry weight within reasonable expectations by endgame. Plus 2x120-ish carry weight for 2 companions (because the mod cuts their carry weight as well). A total reasonable max of 400 pounds.
5. FNV doesn't allow to overload companions, and if you're overloaded, it's next to impossible to play, so it's not an option.
6. In FNV there is no analogue of car, if you want to stash your excess loot somewhere, you will have to return to it yourself to make use of it, which takes too long to be useful.
7. In FNV power armor doesn't give nearly as much benefit as it does in FO1+2, with only +1 STR bonus.
8. In FO1+2 you have 375 max carry weight with all perks and special gear, and 250-350 carry weight within reasonable expectations (that include near 10 STR because you'll obviously be using power armor, get +1 STR implant, and will probably take 1-2 carry weight perks).
9. In FO1 you can have 2 decent humanoid companions (well, at least 1), whose carry weight limit is, well, limitless in practice, due to how the whole companion interface works (or rather doesn't) in FO1. Ergo, you will have no problems with hauling all the shit around, at the low cost of having Ian or Tycho moving at snail's pace, IIRC.
10. In FO2 you can't abuse that trick too much, but:
* you can easily have more than 2 companions
* these companions can wear power armor, making their carry weight at least 250 usually
*
you have a car
Basically, you can always load up your haul monkeys with loot and carry it to the car to store. And the car is almost always very easy to come back to, and it's kinda a given that it travels with you.
Basically, FO1+FO2 = no carry weight problems, unless you're playing a solo wimp. FNV = little to no problems with softcore vanilla and companions, big problems with Jsawyer and companinons, or in vanilla solo hardcore, and absolute nightmare if you go solo Jsawyer HC.