The weight/inventory limit is annoying, yes, but I do feel that if removed it can create imbalances in other aspects of the game. In particular when it comes to the economic function of games. It's there to help deter people who collect ever tiny little cheap thing just to to sell for that extra bit of money. That's the major end result of loot, the game economics.
When you go to loot, most games have a window that pop up showing you whats in the loot and you have the option to determine what to take or not there. A lot of people just click to take everything, or set up their game to autoloot, and then they run into the problem of the weight/inventory limit very quick.
It's not just weight/inventory limit that's annoying, it's loot existing in general that is technically annoying. I can easily see someone ending up with so much loot, that they spend a good hour trying to find shops with enough money left to sell them in.
If you get rid of weight/inventory and all everyone does is autoloot, why even have any loot and inventory at all. What's the use of all the little things in Fallout and Skyrim, the garbage items in Dragon Age that you just end up pressing a button in your inventory to trash anyways.
All that loot is just trashed or sold and are ultimately a waste of the developers time creating if the players don't even care.
Why not just pick up money directly from the enemies killed, forgo the whole annoying loot gathering thing, and use the money to buy what you want from the shops, which would have all the weapons available.
Ultimately, removing weight/inventory limits is dumbing down and over simplifying a RPG game because there are people who want to just blow through the game as quickly as possible.
For me, personally, I like my rpg games to be well over 40 hours, ideally over 100 hours. I like taking my time in those games, I like loot systems, I like finding random books and objects with lore information to sit and read. I see there is a perfectly good reason for weight/inventory limits and again, although annoying, I feel it does play a minor part of the game that has many consequences if removed. I think it all adds to a level of immersion for the world, adds to the role playing aspect of a role playing game, and the slight amount of complexity to it makes the game interesting.