So one of the biggest problems in 5e is how limited Martial type characters (Fighter, Barbarian, and Ranger, to a lesser degree Monk, Rogue, and Paladin) can be both in combat effects (not effectiveness) and out of combat interactions. We see this sort of thing discussed obliquely in things like Spellcasting discussion threads, often discussing how OP Spellcasters are.
5e at it's base "Attempts" to offer alternatives by allowing players to sacrifice attacks from their attack action to initiate grapples or push targets, but very little else that doesn't require a Subclass. Advanced 5e provides each class a list of potential Combat Maneuvers from which they can select a handful over the course of their leveling and have a limited number of uses per turn, creating a new, separate, economy.
What if, instead, we made it so that Combat Maneuvers (Like Grapple or Shove) didn't negate your damage potential from a given attack based on your level?
For example, a Shove attempt at level 1 could deal your Strength Modifier in damage. If you have Extra Attack, 1d6+Strength. If you gain 2 or more attacks from Extra Attack, 1d6+Str+Proficiency Bonus. (I chose 1d6 because Monks deal 1d6 with unarmed strikes at level 5, so it isn't stronger for a monk to spend all their attacks on Shoves)
A Fighter might still miss out on bonus damage from their weapon, but it would certainly make combat maneuvers more attractive. Particularly if we expand those combat maneuvers to cover additional situations, or more accurately use a simple method that can be adapted to other options, such as Dirty Tricks, Disarming targets, or Tripping them as the Player makes suggestions and the DM determines whether it's valid. (No disarming a Dragon of it's claws, for example... unless the DM is cool with lopping off limbs!)
Forced movement, temporary shutdown of Legendary Action options, removing the target's ability to perform reactions, throwing sand in their eyes to give them disadvantage on perception checks and attack rolls for a round... Lots of options for Players to creatively use the Combat Maneuver option, and potentially increase the damage die, effect, or saving throw DC through environmental actions. Such as swinging on a chandelier to "Shove" a target resulting in the target getting pushed farther, taking two dice of damage, or having a +2 higher DC for the Strength Save to resist.
This would put a bit more weight on the shoulders of martial characters who intend to do more than strike their target, but so long as the method of handling the maneuvers is simple (A saving throw or skill check against a fixed DC to avoid either the effect or the damage) it shouldn't be significantly troublesome.
You could even create Feats or Class Features that allow a subclass to explicitly use a Combat Maneuver as a Bonus Action, or possibly reaction.
Out of combat functions are trickier. And should probably be tied to individual class identities. If anyone has suggestions, I'd love to hear them. But I think I might include this general improvement to combat maneuvers in my games going forward, including those involving Advanced 5e/LevelUp play, since it has a "Combat Maneuver DC" built in to make the whole matter easier.
Bonus Points: It encourages Strength Martial Builds for people who want to do a bunch of Combat Maneuvers. Including Strong Monks.