"Is there something you’d like to speak up and add to support the Bard’s arguement?”"
This seems to be requiring the player to be skilled at the negotiation, to know the better ways to support the discussion.
Do you do the same with joint medical checks? Does the player actually have to know a better way to treat "troll blight" than the firstvone used?
It’s a tough question to answer. Both yes and no.
It really comes down to the narrative “weight” placed by the story / the players on the scene. The less important the scene, the more likely its to be resolved with a more abstracted check, while the more important the scene the more detail goes into our descriptions.
So, yes, the negotiation and treating trollblight would be handled the same. I do ask for players engaged in an “important” negotiation scene to — even if not acting in character — to describe their overall intent and approach. What do you want from the NPC? What are you offering to the NPC? And do you have a particular strategy (e.g. drawing on a shared background with NPC, using something the NPC just said to trap them into agreeing, playing a tune to pull on their heartstrings, intimidating them with threat of an approaching monster, etc). In the example I cited, the player said I “Help” without further explanation so I ruled that he could not.
With the treating trollblight...assuming it carries heavy narrative weight...maybe there’s steps involved — identifying that its trollblight, finding patient zero / infected troll, acquiring sample, researching an antidote, brewing antidote, and applying it to the victim. So I’d possibly — depending on specifics, this is all theoretical — expect a PC wanting to Help another with treating trollblight to declare where in that process they are helping, and generally how they’re assisting (soothing patients while samples are collected? managing the distillation & titration process? wrestling an infected troll?).
And no, they’re a bit different by necessity because D&D involves actual dialogue as one of the ways players have fun. Interaction, along with combat & exploration, are pillars of the game. Always have been. No special knowledge is required to do a bit of roleplaying or state your character’s intent & approach. Very possibly I wouldn’t require a roll at all if the player’s position in negotiation was strong enough. In that case there’s no a Persuasion skill, it’s just roleplaying. DM decides whether check is necessary, as per 5e rules. Whereas getting into descriptions of specific skill uses — horseback riding, or wilderness survival, or medieval / Renaissance-era medicine — is not at the heart of the game, it’s not one of the 3 pillars. Certainly, it can add a lot of flavor, and a savvy player might coax a DM into granting advantage for a vivid description, but there’s no expectation built into the game that players are going to, for example, know how to treat a wound, build a shelter, or take a wild horse.