I'd replaced a frozen fan earlier in the summer and haven't heard it come one, but I'm not sure if that was due to actually not needing to (no light popped on the dash). . .
In a correctly working cooling system, the fan starts
before the light is lit. The light indicates overheating; fan activation is designed
to prevent overheating. If the fan doesn't start until
after the overheating light is lit, the system is faulty.
The wiring diagrams I've looked at so far haven't suggested anything, but also seem a little incomplete.
Which diagrams are you using? Do you have online links to them? What about them seems incomplete?
The fan motor is connected to a plug by two wires. That plug
receives two other wires. One is a brown ground wire; the other is a wire that connects directly to temperature sensing unit terminal A2, unless the wiring has been modified. Both Haynes and Clymer wiring diagrams show
that wire as a Violet/yellow wire.
What is showing up at your fan's plug?