Inns and taverns are THE meeting place in a medieval style setting. While there are plenty of other places to start things off, an inn/tavern is the most universally accessible location besides an open, public space. Other locations may often lack any reason for some characters to be there but everyone eats, drinks, and needs shelter. Inns are also incredibly generic and as such can easily be used to set any number of tones for your adventure or campaign setup. 

The Action Start
In this start, the inn becomes a sight of an action scene that instigates the adventure. After a brief setup, the PCs either knowing one another or having introduced themselves, suddenly find themselves in the middle of some sort of action that starts the adventure. Perhaps the man they are all looking for gets wide to them and springs from the crowd to try to get them before they get him. Maybe that same person tries to sneak out and the character spot him. It could be that the action only indirectly addresses the adventure. Perhaps the city guard, minions of an evil NPC, are in the inn, taking prisoners or searching for the PCs themselves, their master having been tipped off to the characters being there to thwart him?

The Mystery Start
The players have come together and find themselves in the midst of soemthing mysterious that sets off the adventure. This is not just a simple quest giver introducing a mystery plot but something mysterious occurring whole the characters are in the inn. Perhaps a specretal presence makes itself knowm, signifying the beginning of something, even the adventure. They might witness a suspicious interaction between inn patrons. Perhaps the lights or flames in the inn act strangely as a hauntng sound filters in through the windows and doors.

The Unintended Quest
As the party gets to know one another, or as they enjoy their latest rest after a long adventure, something occurs within the inn to make them aware of  a possible quest. This can tie in with any of the previously mentioned sorts of starts. In these sorts of starts something such as a rumor, a tvern tale, a song sun by a bard, or even a curiosity among the decorations fo the inn draw the players into an adventure. Perhaps they ask the inkeeper about a strange monster head hanging above the fire. The bard's song might sing of a place that sounds worth finding. An old drunk mutters something about gold in the dark hills. This sort os start does very much require buy-in by the players.

There are a small number of character concepts/tropes that are generally bad ideas for the gaming table, not because they are inherrantly bad characters but because players tend to play them poorly. These character types often appeal to players because in media they are often very interesting characters but what is often lost on potential player. 

Next, the player of such characters should make an effort to express that changes and character growth are occurring. In moments of role-play it only takes a few little expressions to reflect this. Perhaps the shutdown, closed-up character shares a rare and new moment of openess. The evil character might give pause before committing and act that once came so esily to him. The pilfering thief puts the thing back. these moments don't even have to be made in the form of dialog or moments shared with PCs. they can be narrated as downtime activities or in interactions with NPCs and even monsters. A violent, "kill 'em all" sort of character who has spent the adventure launching into needless combat might show mercy or restraint at the start of combat, but this becomes clear when the reason is given.

Problem characters are only problems because people play them poorly. Now, some of this comes down to bad players choosing them for the wrong reasons but, if you are a good player who truly wants to play one of these character archetypes then by all means go for it. Just remember to make it fun, embrace the story that those archetypes are meant to tell and accept that at some point for it to work, the character needs to tramsform.

As I understand it, these sorts of game place an emphasis on collaboration both in terms of a game's plot, story, themes, even worldbuilding with the goal being greater player buy-in and an increased sense of belonging and participation. this can be great and, to a certain extent, I do enjoy and encourage GMs and players to embrace this. However, this is often overplayed or pushed too far. While the intent may be to give players some say in shaping the game and making sure the game is something they want to play, it is possible for the game's balance between players and GM to fail.

The GM is often the player putting the most amount of actual work into the game. Players need only show up and play along. Players who want to express their characters selves should find ways to do so that work within the construct of the game world and not disrupt it. Most GMs are going to be very collaborative and considerate of player's desires and generally find it quite easy to find a way to fit it in or figure out a way to let a player player a character they want, in some manner or another.

At the end of the day, players are in charge of their characters but just as they expect the GM to be responsible for presenting a game that is fun for them, so should they also respect the work, time, and the world the GM is putting before t hem.