The quintessential HL trait/experience is a continuous campaign that doesn't contain cutscenes with action spanning a continuous period of time as if the player is living within the game. That's it, there's nothing suggesting new elements could not be introduced. HL2 introduced an ability to lead small squads of AI resistance fighters, why not transform AI into human controlled?
The franchise has already brought forth games that allow us to play as otherwise tertiary characters such as Barney Calhoun and Adrian Shephard. Why not allow players to play as these characters, not individually in their own campaigns like in the past, but together in cooperation?
When I talk of class system, the characters would have more subtle specialties than a game like TF2, but a character like Adrian Shephard would understandably be a better combatant and be better with typical weapons while Gordon with his suit would be needed to get the group through hazards, etc.
Sorry, but a game like that would destroy a boring rehash of what we've already had with HL1 and HL2. I hope you're not against such an idea because you have no friends so you're terrified of a game built around sharing an experience with friends, that would be tragically sad if true.