Role-selection co-op game?

The Lord of the Rings "Living Card Game" is something along those lines. In each round, a character in the game can participate in combat, questing, or performing other more individual actions. They cannot be involved in all of them, but can certainly change from round to round. That choice is really at the heart of the game, and it is well-implemented there.

That said, the cards definitely have biases so that certain cards are better at combat than questing, for example, but the principle certainly holds. And, overall, I'd consider it to be one of the best-designed cooperative games that I've played.

Pandemic is also a great game, but, as you say, you pick a role and you run with is, as is also true with the Dungeons & Dragons dungeon crawl games (Castle Ravenloft, Wrath of Ashardalon, and Legend of Driz'zt).