While I can understand the decision to move grapple and shove away from its special snowflake "this works unlike anything else in the game", making it an attack roll also produces weird effects, where it's easier to grapple a polar bear (AC 11) than a mastiff (AC 12). The bear is more likely to break free, but because escape rolls are at the end of turn, this doesn't actually matter very much.

I feel like it would be better if both abilities were moved to being a save against your Strength save DC.

I agree that it should be a save. Attack rolls against ac create rng escape difficulty, and attack rolls against a different dc would make no sense.

I vote for attack roll vs strength or dexterity score. So strong and nimble things are resistant, but Monks are great at it.

I think grapple and shove should be Strenght contest.

Save and the target should have advantage if it is one size larger than the individual initiating the grapple or shove

I've been using unarmed strike attack roll vs 10 + the enemy's DEX save modifier, followed by a Strength save (DC = 13 + PB + the ability modifier used in the attack roll.)

The skill contest had to go. It's incredibly easy for players to stack advantage and really high bonuses to ability checks, while monsters generally don't have access to those game features and rarely even have athletics/acrobatics proficiency if they're not a humanoid. Plus the ability checks bypass conditions that should impede attacks (e.g. blinded), bypass Legendary Resistance, and lead to story/mechanics mismatches where characters (including NPCs) that are bad at fighting but min/maxed for Athletics/Acrobatics can outgrapple combat specialists.

You need an attack roll for conditions to work as intended but it can't be against AC because it doesn't make sense for armor to help. DEX saves already represent your ability to avoid things by moving out of the way, so it makes the perfect target. But it can't be an attack roll vs a saving throw, because then things get screwy and one condition can double-punish the target by giving them disadvantage on the save while also giving the attacker advantage. So a "passive DEX save" gives you a mathematically equivalent roll without the double side effects. This is similar to 4e's reflex defense and 3.5e's touch AC.

But targets should also be able to use brute force to resist a grapple, so you need a Strength save too. However now the attacker has to jump through two hoops to succeed, which lowers their odds, so the DC is 5 higher than the standard DC fomula of 8 + PB + mod to compensate.

This avoids the problems with both the 5e and 1D&D rules, while still allowing defenders to use either strength or dexterity, not screwing over monks, not screwing over monsters, and making martial classes good at defending against grapples by default.

Just to be a little bit pedantic, but as far as the play test currently goes, the old grapple and shove should actually still technically work. It has just added an additional way to grapple or shove when landing an unarmed attack. Presumably in the long run, they may be removed as actions but I see nothing in the play test that says the old grapple and shove actions do not work currently.

We are all assuming anyways, without actual certainty in this area. However grapple going unarmed only does make sense, kinda hard to grapple somebody while carrying a long sword and shield, tho shove should still be quiet valid while still wielding a weapon.

I agree the current opposed check and the way characters can max bonuses does make it too easy to lock down single monsters. Perhaps a slight variation on the above. Treat it similarly to a battle maneuver so a hit to AC but a “maneuver” save where a dex based or str based save could be used to avoid the grapple. 

I think it should be a Str/Dex save Vs a DC = the grappler’s passive Athletics score.