The TVTropes Trope Finder is where you can come to ask questions like "Do we have this one?" and "What's the trope about...?" Trying to rediscover a long lost show or other medium but need a little help? Head to Media Finder and try your luck there. Want to propose a new trope? You should be over at You Know, That Thing Where.
In Franklin's case, I think he's just Out of Focus for those missions. In other cases, if they're implied to be actually be doing things off-sceen, I would think Hero of Another Story. If they're implied to be good at what they do but are never shown or mentioned doing anything (even off-screen), maybe The Pirates Who Don't Do Anything. If they have a big reputation but never do anything to support it, maybe Miles Gloriosus
@Beer Baron - several of those are definite possibilities. Out of Focus might be a good one to investigate especially re: Franklin if it isn't already listed. Thanks!
Is there a trope for major characters in a TV series, video game or film series in which they are played up to be major protagonists - but when it comes to missions or major storyline events they go along for the ride but don't really do much? An example from video games if Franklin from Grand Theft Auto V. Despite being one of the three major protagonists of the game, there are numerous missions in which he actually doesn't do much of anything, or if he does, it happens "off camera" and is not actually controlled by the player.