NavmeshAgent vs Character Controller = Absolute Unity Failure?

Well, I will start off with a statement: I searched all the internet a few times and found only partly informations.

The deal is: The collisions between player and NavMeshAgents, of course. More precisely- complete lack of them.

Now, I know, that you can add a NavMeshAgent on the Player as well, which wil make a great effect of best collision detection possible. Great. The only downside is - it prevents the Player from a. falling anywhere and b. jumping.

Now, imagine that I'm making a lightning-fast, Unreal Tournament-like shooter, where player's mobility is based on constant jumping, platforming, dodging and so on.

So from what I researched, I would need to screw the useless now Character Controller and write the Player movement script all from scratch OR try to create all the AI and Navmeshing, again, from scratch OR write a script which will disable NavAgent on jumping, with a side effect of not being able to fall, make the Player phase through enemies while jumping and make all the jump landings choppy,automatically "sticking" The Player to the nearest point on the NavMesh. Herp Derp.

Is there something obvious that I'm missing, or are the basic Unity functions not compatibile with making FPS-Player vs NPC games? And I doubt the second option so please, PLEASE, prove me wrong...