Personally, I use a sort of slot system for equipped items (anything kept on your person rather than stored in a container). You can carry one one-handed weapon or two light weapons on each hip and one two-handed weapon or two one-handed weapons on your back, and you can also carry up to one weapon in each hand, though obviously that leaves you short a free hand. A shield likewise takes up a free hand when equipped. You can also equip one belt pouch, a water skin, and a set of armor.
Where weight comes into play is with items stored in containers. I observe the container weight limits in the PHB, so a backpack for example can hold up to 30 pounds of gear (rations only weight 1 pound each instead of 2), and you can strap up to one additional item to each side, which could be more weapons if you want, but the smart play is usually to strap heavier traveling essentials like your bedroll to the sides.
I find this to be a nice, manageable system for inventory management, but it doesn’t making strength matter. Personally, I haven’t liked the results when I’ve tried using encumbrance for this purpose. It punishes dumping strength rather than rewarding boosting it, which I’m not the biggest fan of. But in my evaluation, the problem of strength not being useful seems to stem from the fact that Dexterity gives characters both an offensive and defensive boost, while strength only really gives an offensive boost. The exception is with heavy-armor wearers, whom I see tend to opt for strength over Dex. So, my solution is to add light and medium armors with strength requirements. I haven’t tested this yet, but I plan to in my next campaign.