Hold up. This is how Shove is supposed to work:
Using the Attack action, you can make a Special melee Attack to shove a creature, either to knock it prone or push it away from you.
Note: If you’re able to make multiple attacks with the Attack action, this Attack replaces ONE of them.
The target must be no more than one size larger than you and must be within your reach. Instead of making an Attack roll, you make a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you win the contest, you either knock the target prone or push it 5 feet away from you.
So it doesn't matter how many Actions you get, you can only Shove once per round. Once again, WotC thought of these things. If they'd just actually implement it, Shove would not be broken in BG3. Again, it was never meant to be that a player, like me currently with my present Barbarian Dwarf playthrough, could go through the game only shoving and throwing and killing everything. You shouldn't be able to Throw an intellect devourer over the railing of the top deck of the nautiloid, sending it flying 20 feet-ish before plummetting to the deck below, and it was certainly never meant to allow me to shove a thrall 10 feet to the railing, another 20 feet across a staircase leading up, and then over THAT railing on the far side of the staircase from the lower deck over the railing of the nautiloid to send a thrall plummeting into the Hells. In other words, I should NOT be able to yeet a fully grown butt human being 30+ feet so that he flies over two railings from an upper deck over a lower deck and off a ship to be a one-hit KO.
If they would implement the proper 5e rules, Shove would be exactly what it was meant to be - a useful combat option for:
1. Shoving creatures on the very edge off the edge.
2. Shoving creatures prone so melee allies can get Advantage on their attack roll.
3. Shoving creatures 5 feet so they are no longer within melee attack range of an ally so that ally doesn't have to disengage. They can, instead, Dash to get away (because the prone person must spend movement to stand, so even if they Dash also they can't catch up), or they can move away and still attack without initiating an Attack of Opportunity.
That's pretty much it. I'm sure there are maybe a few more creative things you could do with it, but being that it should only be 5 feet at most per round (because you can only Shove once), it keeps Shove from becoming, even at higher levels, a tool to shove people endlessly until you push them off cliffs or into lava pits.
As for Throw, the rules state:
Battle often involves pitting your Prowess against that of your foe. Such a Challenge is represented by a contest. This section includes the most Common Contests that require an action in combat: Grappling and Shoving a Creature. The GM can use these Contests as models for improvising others.
So, Larian implementing a Throw system especially for Barbarians, makes sense and is well within the confines of the RAW 5e rules. However, the implementation should be something similar to Grapple.
When you want to grab a creature or wrestle with it, you can use the Attack action to make a Special melee Attack, a grapple. If you’re able to make multiple attacks with the Attack action, this Attack replaces one of them.
So, again, only once per round and it is an Attack Action, not a Bonus (but Larian is already doing that, so at least that's good).
The target of your grapple must be no more than one size larger than you and must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check instead of an Attack roll: a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you succeed, you subject the target to the Grappled condition (see Conditions ). The condition specifies the things that end it, and you can release the target whenever you like (no action required).
Soooo, it is VERY similar to Shove. You make the same exact roll to initiate it, and both should cost an Action so you cannot do both in the same turn UNLESS you get multiple Actions per turn such as with Action Surge or if you are a higher level and get more than one Action per turn. It doesn't say you can either Shove OR Grapple, it just says you can only do each one once per turn. So, it makes sense that at higher levels, or if you're a fighter, you might be able to do both in a single turn by expecting 2 Actions. Thus, again, there's a difference between being a Rogue and a Fighter. If it is an Action, the Fighter can do both in a single turn, but a Rogue can't, which makes sense.
Now, I won't get into the rest of the Grapple rules because we're talking throw here. Notice that it says that you can "release the target whenever you like (no action required)." This would imply that you can Throw something you're grappling without expending an action on the same turn you did the Grapple. Therefore, the Improvised Weapon rules would apply.
This is what the Improvised Weapon Rules state:
Sometimes Characters don’t have their Weapons and have to Attack with whatever is at hand. An Improvised Weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, OR A DEAD GOBLIN.
That would mean that a living goblin should apply as well, right? It then continues to say:
Often, an Improvised Weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the GM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her Proficiency Bonus.
So, you would NOT use Proficiency Bonus when throwing a goblin or other creature. They do not resemble weapons.
An object that bears no resemblance to a weapon deals 1d4 damage (the GM assigns a damage type appropriate to the object).
So, when throwing a monster, bludgeoning type makes the most sense for most creatures since you aren't likely throwing a sharp creature.
If a character uses a ranged weapon to make a melee Attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.
This does NOT differentiate between throwing an axe or throwing a goblin, BUT this calls for common sense. You can certainly throw a 2-3 pound axe further than a 40-45 pound goblin. So, if you can throw an axe 20 feet, and receive disadvantage beyond that to a max range of 60 feet, and a goblin is at least 10 times heavier, you should probably only be able to throw a goblin 5 feet before receiving disadvantage and 10 feet max. I would then also say that it makes sense that you can only throw a creature that is of the same size category as you.
Now, smaller size categories, I could see that for both Shove and Throw, one could implement a tweak to the rules and say that if you are shoving or throwing a size category smaller, you should be able to shove or throw an extra 5 feet. So, human shoves halfling, they go 10 feet instead of 5. Human throws halfling, they can throw them 10 feet to a max of maybe 15. 15 even seems too much to throw a 40-45 pound creature, so 20 feet seems way too far. Tiny creatures could maybe be thrown 20 max or maybe even 25.
I can even see the argument for higher strength, greater distance, but it should be more for creatures/characters with +4 or more strength. Like maybe every +4 of Strength Modifier, you gain an extra 5 feet to Shove distance and Throw distance. So, character with 0 Strength can only shove 5 feet, but character with +4 Strength can shove 10, etc.
That way, creatures with super high strength and large, like ogres, could conceivably throw goblins and halflings and such a considerable distance. Normal throw is maybe 10 feet max. Because they are throwing 2 sizes smaller, they can throw an additional 10 feet for a total of 20 feet. Then, because their strength is +4, they get another 5 feet added for a total of 25 feet (maybe 30). That makes sense. Ogres throwing goblins in barrels 100 feet from the ground up onto a 30 foot high wall, doesn't make a lot of sense. If they were maybe right up to the wall, that would make more sense.
All this to say, once again, that they wouldn't have to get rid of Shove or Throw. They'd just need to tweak the rules a bit that they've already implemented to make them less OP. Nerf distance to 5 feet for shove and make it an Action you can only do once per round. Also add Prone option. Then, for Throw, keep relatively as is but limit the distance you can throw AND apply No Proficiency when attempting to throw one enemy into another so that the Ranged Attack roll is without Proficiency. If beyond a range of 5 feet, Disadvantage should apply to the attack roll so that it is hard to throw a goblin 10 feet into another goblin. Not only would you NOT be proficient, you would gain Disadvantage. Making it possible, but not 90% chance of success like it is now.
Then, if you hit, both take 1d4 damage only. Like an Improvised Weapon, you don't add your Strength or Dex bonus to the roll. 1d4 is it. And if you don't throw an enemy into another enemy, the thrown enemy still only takes 1d4 damage. Bludgeoning.
THAT should fix both mechanics and make them less OP and more of a fun supplement to standard combat mechanics - keeping it well-balanced without all this shoving into lava pits or off nautiloids business.
