Let's face it tracking encumbrance is a huge pain in most RPGs. Tracking weights of items is a bother and gets even more annoying when you break things down to the coin weight. Sure how much things weigh has a serious impact on what folks can lug about with them into dark deep holes but so does volume (don't think so try to carry an unrolled foam mattress through a few door ways and down a staircase?). One of the biggest reasons it is a pain is because aside from the typical impact it has on movement rates set by thresholds of encumbrance it has no regular impact on play.
One way to speed up the tally of encumbrance is the slot method where each item or each significant item takes up a number of "slots" worth of equipment a character can carry. This has cropped up in recent years in OS land every now and then and saw use in The Fantasy Trip decades ago. Of course how big these slots are will vary from game to game and campaign to campaign. They often provide little difference from the threshold impact on movement rates, occasionally a penalty is applied to actions.
So here's an idea I've been kicking about in my own campaign: Save Vs Encumbrance. When a character is doing something exerting, something that could be fatiguing they may find themselves having to make a save vs encumbrance. I use a slot based system myself so the slots of equipment serve as the target number for a d20 saving throw (or ability check). Carrying 10 things worth worrying about? Well climbing that steep slope requires a save vs encumbrance, want to keep running after 5 rounds: save vs encumbrance.
Of course what happens when the save vs encumbrance fails is situational. Sometimes it's just a matter of "not done yet", other times it's a failure, while at others it is a an introduction of a fatigue penalty. The fatigue penalty in my campaign is a -1 to all those things a character has to roll a d20 for. They can keep piling up, i see no reason for a hard limit but it should be clear to a player that bogging their character down with a -10 fatigue penalty probably isn't a good idea.
Players can of course rest or even use stimulants to shake off the impact of fatigue. A 10 minute break shakes off 1 point of fatigue penalty, so does a warm beverage, so does a meal. Each thing required to shake off fatigue has a time cost and in dangerous environments that cost in time can be a hazardous resource expense. After brief rest and a a warm meal the only typical way to shake off fatigue is sleep and that knocks off a HD roll worth of fatigue each 4 hours. Fatigue can be frighteningly enduring but isn't impossible to shake off.
So next time the players want their PCs to swim across a subterranean river in their armor don't worry too much about "swim skills", don't say no, but do have them make a Save Vs Encumbrance and see how that impacts their choices in the future.