I'll add that Pathfinder 2 has a reasonably neat solution, but it is tied to the game's action economy so it would be really hard to implement in 5e without a major overhaul.
In PF2, you have three actions each round. Each action can be an attack, but at -5 on the second attack and -10 on the third (or subsequent if you can pull off shenanigans that will give you more), and since the game is designed so you'll about a 50% chance of hitting equal-level foes (60% for fighters using their primary weapon type) that's a pretty sharp drop-off in effectiveness. Anyone (well, anyone who's any good at the Athletics skill) can make combat maneuvers (shove, trip, grab, disarm), but this counts as an attack and requires either a free hand or a weapon designed to do that kind of thing. This is sometimes still useful – perhaps you want to trip your opponent to nuke their action economy or to set them up for the rogue's sneak attack, or something like that – so you will see situational use of maneuvers.
Fighters (and to some degree other classes, but mainly fighters) can take class feats (essentially selectable class features you get every other level) that build upon these maneuvers. But most of these have something that makes using them non-automatic. For example, Brutish Shove is a level 2 feat that lets you make an attack with a two-handed weapon and if you hit, you automatically Shove the foe 5' back (and can follow them if you want). In this case the limitations are twofold: you have to be using a two-handed weapon, and the feat/action has the Press trait which means it can't be your first attack in the round. Another one is the 4th level Knockdown, which lets you combine an attack with a Trip attempt in a single two-action activity, and your multiple action penalty doesn't increase until afterward (and you don't need a free hand if you have a two-handed weapon). That's strong, but sometimes you'd rather attack twice, or maybe move-attack-move, or something, which means it's not something you do all the time.
Anyhow, PF2 fighters can take lots of feats along those lines if that's what they like in their fighting style. And if they don't, well, regular Trip ain't too bad either.