I do not think it is entirely irrelevant that the 5E rulebook is designed for encounters controlled by a usually somewhat sentient person, the DM, whereas the encounters we are going to face in the game are run by a pocket calculator.
A DM would probably have the courtesy to line up an encounter in a way that doesn't become insanely frustrating for the party, because otherwise the party will start whining like babies in real life, making the DM's life rather miserable. Our pocket calculator AI friend has no such reservations and will happily drop all the chaff right on the squishiest party member as a matter of principle.
And we currently have very little control over the area occupied by the party without using shove. Sure, battlemasters can use a superiority die, we can use one of a handful of arrows of thunder, or maybe we will have a chance to use a one of our limited supply of void bulbs with an action before the position of the party becomes completely entangled with the horde of enemies. Or maybe we have a scroll of thunderwave. Or maybe we have someone who has a cast of spell slot of thunderwave. There are options for those who know where to look, but those are fairly exhaustible resources that it probably wouldn't be good game design to assume a party has a sufficient supply of.
And the enemy horde will always have way more actions than the party. And they can jump. And sprint. And use cunning action disengage or cunning action dash, or just plain dash and action surge. Or just tank that one or two AoO reactions, if necessary. But they can absolutely engage the group and neutralize any ranged characters in the group.
The concept of tanking with whomever has tank duty and using that to occupy the enemy chaff while the damage dealers neutralize the bigger threats is a very common element of party based RPGs, but that only works if the tank can actually occupy or slow down enemies. This is not the case in BG3, in my so far very limited experience. The enemies can and happily will ignore the tank and just flood towards the squishies, and there is just not a whole lot the tank can do about it.
Shove as a bonus action isn't pretty, but it does give the tank more of a zone of control. The tank can now use an action to either control space or deal damage, use a bonus action to have a decent chance to further control the space, and then use positioning to try and hold another hostile through the presense of threat and AoO. And the squishy characters can then actually be squishy characters and still have some chance to be productive.
In my view, shove as a bonus action is fine IF it is reduced to just being a 5-6 foot thing unless the shover has an overwhelming strength and size advantage on the shovee. Let ogres and minos throw party members around, why not? Obviously that doesn't mean 6 foot lateral movement per 4 feet drop, like it is currently. And to fix shieldmaster perk, probably give shieldmasters a +2 to their shove DC.
Shove sending targets fully prone at the cost of an action does not sound entirely terrible, but the AI could then do that to the player party as well. And the enemy horde has probably 6 or 8 or 10 actions per full turn, and most of the companion NPCs are not all that shove resistant. How would a player really combat this, save for more cheesecake tactics?
If they follow 5E, melee types can use one attack (after getting extra attack) to shove the enemy rather than the full action. If the character is a fighter, (and if they allow shove to be two different tiles, one for 5ft shove one for prone) they can get nasty. Ex: Shove prone, attack with advantage, action surge, and two more attacks with advantage. To your point, though, so can the enemies.
I haven’t had an issue with my ranged characters being overwhelmed in melee. If you position your characters correctly you can take advantage of the same things the AI does. Shove needs to be fixed, no question.
As far as it being a BA, I think it’s too OP for both sides. If you face (or are) a thief Rogue you can dash twice and shove. I’ve had a gnoll use dash as an action, move 60 ft, and launch my main melee off a cliff for almost all his HP. This wouldn’t and shouldn’t be possible at the table. Granted, we can do this too. I just think it’s too much for a BA.