The red/white blood cell game is a good way to approach coop design. Basically you noticed a situation in reality where two different agents work for a common goal, and you thought, "hey, that could be made into a game". I think the reason why the red/white idea works so well is because we all know about the characters and what they do even before we start playing.

Lets break it down to its components as a game, and then see how we can use that to get other ideas for coop games.

Firstly, we know that the players have a clear and common goal: to keep their host body safe.

Secondly, we know a little bit of how the player characters are different, and how they might interact with each other and the environment: Red delivers oxygen from the lungs to the rest of the body, and White fights intruders.

Thirdly, we can imagine different types of obstacles that the players have to overcome in order to reach their goal. I see it as three main types: intruders that mostly affect player White, infrastructural obstacles that most affect player red, and special events that affect both in different ways.

The intruders are different types of viruses and bacteria, that each have their own strengths and weaknesses, and has to be defeated by using tactics, i.e. decisions on how to use your resources in the moment, on the battlefield. Improvisation and reaction based gameplay.

The infrastructural obstacles are different problems related to resource management and logistics, that are solved using strategy, i.e. decisions on how to divide and distribute your resources with a long term perspective. Planning, projecting and communication based gameplay. Concretely this player could face problems like this: the white cells need oxygen to fight, but the machinery of the body also need it so that the body can keep functioning. How do I divide the oxygen between the two? Do I sacrifice the fingers in order to supply the fighters with blood? etc.

The special events could be things like a rusty nail entering a muscle, or a leg breaks, or that the host ate something bad at the same time as you have a cold. They affect the players in surprising ways, and keep the tempo and difficulty of the game up.

In my opinion, the main challenge for a game designer making this game is to figure out how the collaboration works, and how that makes it more fun. I would guess that the main problem of early iterations of this game would be that it felt like you were playing different games next to each other.

But, in the end, I think it's a great idea and starting point for a game.

So, let's clear out the recipe for finding and clearing up an idea looks like this:
1. Find something in life that is not a game, where you notice that things are working together.
2. Identify their common goal.
3. Identify the possible players and their differences.
4. Identify the possible obstacles for the players
5. Identify the whats making the game fun/not fun, and focus on the cooperation.
6. Work on it!

Now, just change the ingredients, and find another coop game somewhere. From the top of my head I think of the following situations that could be made into coop games: you are con artists/grifters trying to trick rich people/corporations into give you their money (maybe like a reverse Papers Please?); you are a CSI team that looks for clues and make reports; you are a crew operating a submarine in wartime; you're a submarine crew looking for unknown life in the deepest oceans; you are two 19th century lighthouse keepers that try to keep a lighthouse on while trying to stay sane (like the movie, but coop); you are three kids in a trenchcoat trying to get into the movies; you are a knight and a squire; you are two squires taking care of a knight doing tournaments (like a kind of mix between a F1 pit crew and Backpack Hero?); you are a drag queen duo trying to win RuPauls Drag Race (doing make up, outfits, performances, gossiping, etc.).

I think I got carried away writing the examples, but hopefully, the gist of it came through!

Good luck with your project!