It's compatible in that it is now competitive with other classes.
It already is competitive with other classes. (See below)
You know what fighters can do at level 11? They can attack SIX TIMES in one round of combat.
Once per short rest, not all the time, otherwise it is three (or four with TWF). And you know what? Monks can also attack 3 times every round... at level 5, just like fighters. Games rarely get to level 11 for the vast majority of players.
By level 7, a spell caster can turn a roc into a chicken, just like that! A level 10 bear totem barbarian has something like 200 effective hit points!
Sure, and monks can do things they can't. It isn't a matter of comparing apples to apples, it is about the balance and competiveness of apples to oranges.
Monks in 5e could already run across water and up walls at level 9. Being able to carry someone else is awesome, but it doesn't break the game.
First, I never said
anything about "breaking the game", so please don't put words in my posts that aren't there. Being OP =/= "game breaking."
Anyway, but this:
for example, in a recent battle she picked up the barbarian, carried him 100' across a moat and up a wall to tank for her, and still got two attacks off on the enemy spell caster, including a stunning strike, while putting hand of harm on his bodyguard to leave him automatically poisoned.
Sadly, now that I really think about it, this was already possible in 5E:
At level 10 monks already have a speed 50 (given 30 base), so move and step of the wind to dash = 100 feet. picking up the barbarian (since they are compliant), is
arguably a free object interaction (a stretch, but
shrug), making two attacks is just the Attack action. Stunning strike is available already, as well. And with Tasha's (OP...
cough cough), hands of harm is already a thing, too.
I'm not sure about the Strength of your monk to pick up a barbarian and how that should have affected your speed, depends on what, if anything, your group does about encumbrance.
Now, the above would burn 2 ki, sure, but given the situation and how awesome you found it, in 5E burning 2 ki for how "really good" the monk is, it seems like an appropriate time to burn the 2 out of possible 10 ki...
So, really, what did you do that isn't already in 5E that made that scene so great??
I recall one adventure where a player's monk PC leaped over a 30 foot chasm after the BBEG cut the rope bridge, landed next to the BBEG who was holding a captive hostage, hit the BBEG with a successful stun, and then used a shove "attack" to push them off the edge into the chasm below, effectively killing the BBEG. And this was around 6th level IIRC. Still really cool, and nothing any other class could have done. Needless to say, the DM was a bit shocked, miffed, etc., but laughed it all off as a really awesome "hero moment" that tales were told of later on when the captive told the people of the city about it.