Everybody has their own tolerance levels for how much paperwork in an RPG they feel is fun. For some people, updating their character sheet is a milestone system for their character. It's as much a part of the game as NPC interactions, decisions, strategy, solving puzzles, and so on. For others, a character sheet is a perfunctory obligation performed at the start of a game, and only begrudgingly updated. In either case, it seems that one of the most laborious tasks is calculating a character's limitations for carrying loot. Not only are you meant to write down an item you've acquired during a game session, but you're supposed to write down the item's weight, add it to the total, but you're also meant to remove items and weight when you sell or use an item. It's a fair bit of work, and if you're using paper then it's a lot of erasing.
I acknowledge this, and I don't think tracking weight is necessarily for every game. But in some cases, me and my players get a lot out of enforcing encumbrance rules.
When every item you pick up weighs nothing, the value of every item is essentially equal. There are minor differences that may make one item more valuable than another to a player, such as the amount of damage a weapon deals. Aside from that, though, ignoring weight makes all loot the same. It's some imaginary stuff you throw into an imaginary bag and then forget about.
This has some severe effects on your game.
First of all, you should never feel that loot is imaginary. Obviously we all know everything in the game is imaginary, but you still feel anxiety when a big creature pops out of the shadows. Likewise, you should earnestly feel excited when you find loot. You should feel anxiety when you have to decide whether to drop the gemstone you found earlier so you can pick up the gold-encrusted skull.
Just as importantly, you should feel a sense of accomplishment when you have just the right item at just the perfect moment. And you should feel frustration when you had the right item, but left it behind because you were carrying too much and really really wanted that gold-encrusted skull instead.
These are game moments you don't get when there's no carry limit. When you're able to collect everything you ever encounter, there's no decision to make. And when there's no decision to make, you can't you look back at the path you've taken and reflect on how clever you were for arriving at this moment.
If you did have to consider how much stuff you could carry, what would be the deciding factor for what you picked up and what you left behind? You might consider damage amount and type for weapons, anticipated monetary value at the next market you encounter, significance of the item to an important NPC, the history of the item, the aesthetics of an item, the usefulness of an item in a specific circumstance or a broad set of circumstances, and so on.
Some of these things might come up even without you having to make a decision about an item you've uncovered, but not most of them. The "right" thing to do when you have no limits on what you can take is to just take everything Throw it in the bottomless bag, move on. You don't have to ask about the lore behind an item, you don't have to assess the item's value. If it's important, the game master will tell you.
Strength is often an important general stat. It's not usually a first pick for a dump stat, but depending on the character build, it's sometimes a contender. After all, if you're building a wizard or a ranged attacker, do you really need strength?
Well, you might if you have to worry about how much you carry. Part of the game in an RPG is the choice of how to "spend" your stat budget. Not all stats can be good. You have to choose a stat to starve in order to bolster another. The less a stat matters in practise, though, the more it becomes the obvious and painless dump stat. Choosing which stat you're willing to live with as your disadvantage is part of the game. It's not "fun" in itself, but it helps make your character interesting and unique.
Encumbrance ensures that sacrificing strength actually means something.
A common argument against encumbrance is that players are going to end up getting a bag of holding anyway.
If that's true, then I agree that encumbrance doesn't matter. It's happened in games I run. But there are some important things to consider.
First of all, players don't have to find a bag of holding. If the adventure you're running doesn't provide a bag of holding, then you're not obligated to throw one in. There's no law that a vendor has to have a bag of holding to sell to them.
And when an adventure does provide a bag of holding, there's no guarantee that your players are going to have that bag forever. Somebody has to be carrying the bag of holding, and should that character fall into a pit of lava or get abandoned in the Abyss, then the rest of the party needs to understand that they no longer have a bottomless bag in which to stash stuff.
A bag of holding is not a foregone conclusion. And that's how it should be. A bag of holding isn't special when you get issued one in every game.
A bag of holding is a powerful and mysterious magic item, so make it rare.
I don't think encumbrance is right for every game. I wouldn't track encumbrance for a short game, or even during every long-term game. But it's a great tool to have as game element that reinforces both mechanics and storytelling. Consider trying it in your next game.