New Radiator, Computer Won’t Turn On

An AOI Cooler will not have any effect on booting up the computer if it's plugged into the cpu fan header.
More than likely, you disconnected or loosened another connection and did not realize it. Go back through and check that all wires are tightly connected. I would remove the CPU also and look at the pins in the motherboard socket to make sure you didn't damage them while installing the cooler.
If that fails, remove everything from the motherboard except CPU, Power Supply, and Ram (reseat these just in case) and try to boot.
If this still fails, you most likely damaged something while installing it.
 


What could i have damaged, though? No bent pins on the CPU, unplugging GPU and storages from mobo did nothing. Can you damage a mobo that easily?
 
The easiest thing on a motherboard to damage are the cpu pins.
If you smoosh the cpu around to get it seated the socket, you have likely damaged some pins.
It is meant to be dropped in.

Intel sockets are certified for a maximum of 15 insertions.
You do not want to remove a cpu from it's socket once it is in.
 



Was the system unplugged from the wall, and the power button hit to drain the leftover juice in the motherboard?
If not, it could be very easy to damage your board from any static shock. Also, as for the cpu (SOCKET) pins (NOT THE CPU ITSELF) if you pushed down too hard on the AIO when installing, you could have bent the CPU socket pins in the process.

To correctly diagnose, Take the motherboard completely out the case and set it on a cardboard box. Connect the Power Supply and insert the CPU and try to start the system as-is (No ram installed). You can power it on by touching a screwdriver to the motherboard pins corresponding to the front panel power button.
You should get 4 beeps if you have a pc speaker on the board.
If you get the beeps, you can try to add 1 stick of ram and test. If you don't get the beeps, more than likely your board, power supply, or cpu is dead. To test any further from this point, you need replacement board, PSU, and CPU to test with your stuff to find a culprit.