there's nothing in the rules telling you things you're holding don't get helped by slow fall.
So to get this out of the way, rules as written say absolutely nothing about when you stop falling if someone catches you, much less how to catch someone that's falling.
We will just skip to the part where the monk and lidda are falling to their deaths as one mass. Our monk buddy is going to cancel the fall damage just fine due to his class feature (though one could argue he needs a free hand or something to touch a wall depending on the world). Lidda also gets off easy, so long as the monk keeps a hold of her. From a rules perspective, anything you're holding is usually part of your inventory and the class feature includes the halfling our monk is holding. I certainly don't see any part of the rules demanding that you make saves against the thousand gold pieces in your pockets ripping the muscle off your skeleton when you hit the floor, so a halfling is no different in this case.
Of course, more realistic GMs are welcome to give the monk a pass, and hope the halfling doesn't rip his arms off when they land. She is an object with mass and momentum, so if it's your idea of "cool story bro" for the monk to potentially lose an arm knock yourself out. There's nothing telling you no either way.
Telegraph your decisions excessively, and allow retcons
In my experience, what the DM is thinking is very different from what the player is experiencing. In the example you gave, while you "already knew the weight limit," the player did not. From his perspective, then, you randomly collapsed the platform.
I deal with this issue by not only telling the players what the characters see, but also what the characters judge. For example, instead of telling your player "you see a crumbling platform," you can tell your player "you think the platform might collapse at any moment". In "real life," the characters have a wide range of experiences and perceptions that you can't possibly convey in a reasonable time as a DM, so telling the players what their characters think is an easy shorthand--the players are always free to disagree or probe more deeply, if they choose.
It's always going to be a judgment call on whether you should call for some knowledge-based roll or how much information to give, but I would err on the side of giving more information, to avoid instances like the example above.
Additionally, I let my characters do minor retcons. When your player decides to go on the ledge anyway, you can say something like "the ledge feels like it's about to give way under your feet". If the player changes their mind about going on the ledge, I let them do it.
It's going to be annoying, and it will feel like you're giving everything away, but it ultimately leaves your players feeling like they have a much better understanding of the situation and the logic you're using. Moreover, there might be some situations where the players will want to go on the crumbling platform--in those cases, they will have a good idea of what they're getting into.
"Gotcha" moments suck: focus more on avoiding them and less on mechanical surprises
Here's the thing: even though you're literally pitting the players against adversaries, the DM-player relationship in 5e should not be adversarial. I put in terrain and traps so that my players can have fun defeating them, not so that I can spring surprises on them and laugh evilly.
Even for actual, hidden traps, gotcha surprises are terrible. From the player's perspective, they get put into difficult, damaging situations totally out of the blue. Indeed, when I play traps, I give the players a few seconds to try to respond to a trap activation, just so they have a bit of agency ("you hear a click when you open the door, what do you do?"). By boosting player agency and letting them understand what you're thinking, you can reduce the number of gotchas and likely reduce your perceived unfairness.
Reset your player's expectations
Now, in order to implement this new DMing style, you have to fix your relationship with your players. After all, this problem is not only coming from you, but from your players as well.
You should talk to your players, tell them that you're going to change how you're DMing, and see if they're willing to reduce their combativeness. Hopefully, this "reset" will help your future sessions go more smoothly. Again, you can see this as part of making your intentions more transparent and more explicit, and hopefully your players will appreciate that.
Best Answer
The problem is this part of the falling rules:
You are suggesting you give the PC advantage on the attack and no damage when they really should be landing prone after taking damage.
The question then is whether negating the damage is fair or not. The problem with the falling rules is they don't consider what you land on. If you fall 30 feet and land in a deep pool of water then your damage should be reduced or negated. For this example it's a matter of how you land in the water. A clean dive or a pin drop would be the ideal way of landing, a belly flop would be the worst.
Landing on an opponent would follow similar logic. Unless they are dropping on something hard like a golem their opponents body should be usable to break their fall a bit.
I'd suggest making the PC roll an acrobatics check to land on their opponent correctly. Have the DC determined by distance fallen. If they fail then they and their opponent take damage and they don't get their attack.
Do keep in mind the distance between the attacker and the target. If they are 15 feet up in the air on a branch and their opponent is 6 foot tall then that is only 9 feet.
If they drop too far you might want to consider a roll (athletics maybe) to jump on their opponent accurately.