When other people say that they want team deathmatch or even capture the flag in Brink, I say NO. I don't think that Brink is the type of game to have those game types, it's too team based, too objective based, too different than the rest, that's why I like Brink.
But THIS, this is a great idea, I like how it's different but at the same time, doesn't ruin that teamwork objective gameplay that Brink is. Bravo sir, I have been thinking of making a gametype in which both teams attack AND defend, but couldn't quite put my finger on it. Good idea.
Combat and Goal are separate gameplay systems.
Goal is the game mode, BRINK currently uses Assault game mode (see: unreal tournament)
Combat is a system all its own.
In BRINK, the game mode consists primarily of class-exclusive button-pressing to advance. However, the way this is done is more interesting to perform and varied for each class. it's still like saying "red guy must press red button by holding action for 3 seconds" followed by "blue guy must press this blue button by holding action for 3 seconds" followed by victory. The occasional A-to-B item transportation for all classes as well. The decoration surrounding this simple mechanic is what makes it far more interesting, seem more varied, and fit the story.
Combat systems are a separate matter. "blue man holding action to interact with red man" is not the same as "blue man pressing blue button". In team deathmatch, you would basically have what you already do if objectives couldn't be completed. People would rush towards the enemy spawn and shoot whoever they see while playing in a style they prefer. Teamwork exists without holding action to interact with others, but holding action makes teamwork interactions seem more meaningful and easier to notice (see below for examples of indirect teamwork) Teammate interactions have nothing to do with the objective and work in any game mode you can imagine. Keep all 4 classes and the teamwork is there.
indirect teamwork examples: If you simply have assault rifle and grenades for 2 players in call of duty 2, you could leapfrog ( leapfrog is... one player stays still with ironsights to keep perfect accuracy and covers the other player who sprints forward to advance. The sprinter stops at cover and the coverer becomes the new sprinter. rinse and repeat for infinite safety due to teammate's cover) which uses pure teamwork without ever holding action to interact with each other. Also, one player could throw a grenade into an area while staying away from it. After it explodes, clearing the target area, a rushing player who was standing by could rush in to clear out the area further while the grenadier covers from afar the best they can.
EDIT:
Not bad, although when I read the title I thought we were talking about a game mode where there was one moving objective (a repair bot for example) and the team that controlled it moved it toward their goal but if the other team took control then it would move in the other direction toward an opposing goal.
This is also an option that I thought of. However, you'd have to either have all hacking/repairing/explosive objectives or have a robot that can be transported towards either base.