I just tested theory of no exhaust fans in my case (Silencio S400) and conclusion is that exhaust fans are good. Here are temps with exhaust fans stopped and with exhaust fans working (I ran prime95 small FFTs and Unigine Tropics for 20 minutes and only care about maximum temperatures):
CPU Package 78C - 73C
CPU Core 78C - 73C
PCH 48C - 46C
VRM MOS 76C -73C
PCIe x16 48C - 46C
SSD (OS SSD) 38C - 37C
GPU Diode 67C - 65C
GPU VRM 1 64C - 62C
Exhaust fans helped to me, I guess its a good idea to use them. What we can take away from doyll's assertion is that intake fans are more valuable than exhaust fans, but you need to create some flow in case to make it cool well. Intake fans are more valuable likely because they push cold air into case and cold air is more dense and less viscous. Fans are also able to create far more pressure when they exhaust air, rather than suck it it. You have active pressure and static pressure then. In intake active pressure is lower and you rely on static pressure more. Computer fans are made to create airflow and they are either way weak at creating pressures. They are not compressors, they are not sealed and and unsurprisingly they don't produce much pressure. If I understand correctly, computer fans are able to take in air and then exhaust it at fast rate. They have just enough pressure to deal with some obstructions, static pressure fans have better ability to to deal with them, meanwhile airflow fans suffer greatly with any obstruction at all. The main point in fans is to have good air exhaust velocity and just enough static pressure to deal with obstructions, else you need more rpms and lose some velocity.
I think that my first advice was a little bit shit. But when I thought more, you need to put 2 static pressure fans on radiators. You are either way advised to use static pressure fans as intakes to beat resistance of grilles and other obstructions and you should use static pressure fans on radiator anyway, so it saves fans. You can install AIO as bottom intake (but consider that it's AIO and I have no idea if it affects longevity of it) and then use stock case fans as two top exhausts. That should create a nice air chimney effect. It should work rather well, despite GPU mounted horizontally. Here's an example of chimney airflow:
That's probably the best bet with least amount of fans used. Your main problem is that 280X doesn't have any proper rear exhaust and you only have top exhaust. So in typical built, people would use front intake and top exhaust, but air doesn't like bending 90 degrees. As for grilles, you can likely remove top grille without dust consequences as you still have glass cover for dust to land on it. Bottom intake doesn't have tall feet, so maybe filter is a good idea. If feet were tall, you could use it without filter as fans only have so much force to suck dust from floor. ANd if you have static pressure fans, they will have to spin faster to create same airflow as generic fan, so be warned. Anyway, just test out various things and see what happens, theorizing so much isn't good without putting ideas to practice