I know not every table goes about these things the same way but here's how we do it in our 5e game which seems to have the benefit of avoiding the situations presented in the video.
DM sets up the scene in the tavern and the party believes the bartender may have some important info.
Player 1: My PC strides confidently up to the bar and says to the bartender "I hear you might have some information for me."
At this point, the player would not just roll a die unbidden. Instead, the DM would determine if the character's stated action has a chance of success or failure and if there is a meaningful consequence to a failure. I mean, the words of the PC in this first example seem a bit vague so the DM might roleplay the bartender to elicit some clarification about the PC's approach and goal - and have some fun with the scene and hint at the attitude and/or motivations of the bartender.
DM [roleplaying the bartender]: "About...?" or "What's it worth to you?" or "Why, hells yeah, I have information! We have a drink special on mead tonight... buy two pints, get the third free!"
Assuming the player then clarifies their intent, the DM can assess and decide if a roll is needed or not. If needed, we often find it helpful to provide the player with the DC
or tell them it will be an opposed roll. We also find it helpful to define the stakes which could be binary or could scale with the magnitude of success or failure, depending on the situation. In this case, it seems the adventure relies on the PCs coming away with something here so, at worst, they will glean at least an obscure and/or dangerous lead for tracking more info about the Black Coin.
DM [determing a roll is necessary]: Ok, this will be a CHA(Persuasion) roll opposed by the bartender's WIS(Insight). Succeed and the bartender is going to provide some information to you, fail and she is likely going to think less of you and your friends and subsequent attempts to ply information may be more difficult. (or some such thing)
Unlike the video, at our table:
- the DM is not suggesting things for the players to do, as he does with the last PC. The players come up with what they want their charcters to do on their own.
- the players aren't just rolling to do the thing; they are providing a goal (e.g. to get info about the Black Coin) and an approach (e.g. appeal to the bartender's desire for gold); the DM determines if a roll is needed
- the DM doesn't call for a roll when the failure state is "nothing happens" like for the second PC who wants to notice something about the bar. Maybe a WIS(Perception) check would be appropriate with a failure state of the bartender getting anxious that the ranger is casing the joint and a success meaning an empty tip jar is noticed behind a bunch of dirty tankards... leading the PCs to perhaps remedy that situation and gain advantage, or even auto-success, on their next interaction with the bartender
- rollplaying doesn't have to be first person; a third person description that is reasonably specific is plenty