[mongocanfly]

Anything electrical can go bad....
1st I would make sure the fan is good...if the fan ground is good then hook 12v to the red wire from the fan...it should run..

2nd i assume the big yellow wire at the relay is your 12v fan supply from the battery...make sure there is 12v on that wire..I think that's what your saying you tested with your circuit tester..if that's good then move on

Next check the signal wires to the relay...I can see some writing on some of the wires..
Does the gray wire say (sender)?
What does the orange wire say?...
I'm again assuming the gray wire goes to the sender
And the orange wire goes to your key switch on 12v power

Is the gray wire the one you say is burnt? If it is I'd replace the sender and the wire

Then next I'd replace the relay if the sender doesn't fix ix

Do you know the brand of the fan and relay?
There should be a fuse somewhere in there on the big yellow or big red wire..