I actually did think about this a while back, and actually built a prototype for the concept to get it setup.
A lot of the attempts at this concept are either extremely gimmicky or require you play one specific class. The idea that I had was pretty simple: You can choose from any class, resupply rooms function as checkpoints, and you have to accomplish a few objectives that are no different from multiplayer TF2, such as capturing/defending control points, moving payloads, etc. Potentially some stages would be just you going from point A to point B. Nothing fancy, just straight play the game against bots throughout a journey across the badlands.