Well, I got some things done:
I had to take the PSU apart to get more info on its fan. It has the part number MGA4012ZB-A15, and according to some Aliexpress listings, it is a 20 cfm unit. The Noctua 40 mm ones give about 5 cfm if I'm not mistaken, so I assume swapping the fan will result in worse PSU temps.
I also got the two 80 mm case fans installed. I made a extension cable with the matching fan connector and wired both fans in parallel.
I tapped the fan mounting holes with a M5 tap, and used M5x8 socket head bolts to fix them to the case. The bolts have a big, grippy head that can be turned easily by hand, without the need for a wrench or screwdriver that wouldn't fit inside the case with all the other parts installed.
As @jmsantos requested, I tested the CPU fan as intake and exhaust. Intake (default configuration) wins by about 5 degrees when running the Prime95 blend torture test, at full fan speeds.
Now the case fan results. At 30% (these are not PWM, but voltage controlled fans), everything is 5 to 8 ºC cooler, and I mean everything, including the mobo VRMs and the SSD, and they don't make much noise at all. At full speed (3300 rpm) however, they improve cooling a lot. For example, without the case fans, while running the small FFT test, my 4670k uses almost 100 W and heats up to 98 ºC, with the Noctua cooler at full speed. With the case fans it only reaches 85 ºC. The system is very loud if it gets to this, but now I should be able to push the processor to 4.0 GHz and not overheat it when doing some CPU heavy work, and I can leave the room while the computer does its thing.