Context

Most of the time, Loot distribution is Players' and PCs' problem. From my point of view, I'm responsible for putting the chest with X loot in the dungeon. The PCs being able to find it and how they are going to handle it should be their problem.

This works in organized play (i.e. AL), since there are specific rules for loot distribution. It also works when your players aren't selfish and aren't playing a selfish character while having My Guy Syndrome. The thing is: sometimes problems happen.

1. Sometimes a published adventure that we're running in a home-game
   (meaning without AL rules) happens to have an incredible useful item
   for more than one character.
2. Sometimes I suck as a DM and don't think carefully about the loot
   I'm putting in the chest (usually happens when I roll for loot
   instead of tailoring it).

The problem

When one of these situations happen, i.e. when things go bad and the loot isn't perfectly tailored for the party easily taking it, what can (or should?) I do, as the DM? I will repeat myself: I want to intervene as little as possible in PCs' actions, and sharing the loot is a PC action, but sometimes the game simply needs this intervention – either because some players start feeling uncomfortable arguing for loot or because the loot sharing takes more time than the fight to get that loot.

In particular, my current party, with some new players, running LMoP, are having some problems with that.

Some recent examples

• Party for six players finds a chest with some loads of gold, one item that is only a valuable (= gold) and 3 potions of healing. They spend some time arguing about the valuable until I tell them it's just gold coins in a different shape. Then they proceed to argue about the potions of healing. I end up throwing in 3 spell scrolls (we have 3 spellcasters and 3 martial classes) so it becomes easier to divide – scrolls for casters that actually can use it, potions for everyone else.
• Same group. Barbarian and Paladin. Weapon +1, both can use, both would be happy with it. They start arguing and going into a min-max discussion, which was sad to hear since they aren't min-maxers and the math was off, but anyway… significant amount of time lost. I wanted to proceed with the session so I just made them roll for it.

I'll be the first to say that my solutions in both scenarios were horrible, but I was thinking more on "I just want to keep the session going" than solving the loot problem, to be fair.

I would note that every discussion is friendly and nobody gets personally offended. In the end, we're all having fun, but I personally don't like the idea of spending so much time in something that should be more trivial, and some players already told me they would rather having me handle the loot distribution.

This means we're somehow not agreeing about how the loot system is handled – I think this should be players' responsibility, some of them would prefer me handling it. This is not a problem large enough to say we're playing different games and the table should be changed, though.

As an additional context/information, we did not discuss loot distribution in Session 0, which was probably dumb in retrospect.

Question

So, again, what can I do? Is there a way to find compromise between both approaches? As a possible solution, how can I approach the players and convince them to form a social contract about loot, which probably should have been done in session 0? (I.e., I'm predicting "talk to the players" answers and I'm already asking for some advice on how to approach this "talk")

Note on loot systems

I'm very used to MMORPGs and loot systems used by guilds, as well as some loot systems for TTRPGs. That's not the point on this question. I don't want to implement a loot system. Best case scenario is suggesting the loot system for the players and let them do what they want.

• Answers simply presenting a loot system won't help me much.

Related questions (and why they don't solve the problem)

This question is very similar, and closely related. Contrary to mine, the DM is interested in intervening, though. A good amount of answers point out my view – that this is not DM's territory to rule over. In particular, the accepted answer is about a loot system – it might be worth mentioning it to the players and if they agree to use it, fine for me, but this is not the scope of my question. SSD's answer is easily my favourite answer in that question and I feel it would be amazing for more important items. The problem with that is that a) I don't want to drive a story about a dispute over a gold statuette worth 50 gp or 3 potions of healing and b) If the situation becomes too frequent (and it seems it will) it will get annoying to roleplay these disputes every time.

This question again gets answers focused on how the loot distribution is Players' territory, not DM's. This is already a premise of my question, so it doesn't help much.

There are some questions about Players/PCs getting items that aren't even useful for them or supposed to be theirs, as in here and here. Thankfully, we don't have this problem – if the item is clearly more useful for a specific character, everyone is nice enough to let them have it. We also have questions on players feeling bad about loot, but again, this is not a problem here. My problem is mostly time and players not willing to argue, but kinda having to (since they don't just want to give away an item that's useful for them).

The Group

Additional info asked by @KorvinStarmast that might make a difference in the answers: Our group is formed by (close) friends of mine. One of the players is also friend with everyone. Other than him, 3 players are from a group and 2 from another. The people from different groups don't have any personal problem with each other. Age goes from 18 to 25. There's no hierarchy involved (one being boss of another or anything like that), but we have a couple and two girls that live together. As I mentioned, the discussions haven't scaled to anything personal (yet, and I don't think they will).