since you have mechanical abilities.. pull all three fan relays. use 2 small screws drivers to pop the covers.. you may be able to inspect the contacts. the springs come off easily with a probe. the braided scrap can be forced over the edge and the movable plate lifted out to burnish the contact faces.. the 5 pin relay uses both normally open and normally closed contact faces..

the 4 pin relays only use the normally closed.

any issues can cause the fans to not operate..

please do not try to manually control the relays by squeezing them. its an odd set up.. get it wrong and you blow fuses.

replacing all three relays is a far better idea..

you may want to with a cold engine. remove the radiator cap.. place a piece of saran wrap over the filler neck.. push a depression into the saran wrap.. seal it in place with a rubber band.. squeeze a hose to verify a tight seal.. you may have to pinch closed the overflow hose.. start the cold engine for 15 seconds.. if the depression has inflated you could have a combustion leak into the cooling system.. leaking / blown head gaskets, cracked combustion chamber. this is only a test..

it takes a scan tool to command the low speed fans on and the high speed fans on.. low speed fans places them in series.. high speed fans places them in parallel. almost all carmakers use this circuit except for some fords that use a fan braking relay stashed on the other side of the engine compartment that prevents the fans from freewheeling in the airflow at speed. people are going WTF.. yep.. takes the circuit going to ground and diverts the ground side of the motors back into the low speed resistor creating a regenerative braking sort of circuit like is used on locomotives.