Cooling Fan won't start on 97 SC2 -- advice on what to check next?

So, on my mother's 1997 SC2, the radiator cooling fan won't turn on and
the car therefore runs too hot if it sits in traffic. It will get to
over 3/4 and the cooling fan would apparently never turn on.

First, I replaced the Temp Sensor (turned out the old one in there was
already the brass type and looked fine). No change. The temp readout
on the dashboard also seems to be working normally.

I checked the fuse and the relay -- both good, no change when I swapped
them. Tried turning on the AC, still the fan did not start.

So, I thought it was the fan motor. But when I tested the connector to
the motor, realized no voltage was being sent to the fan. At this
point, the dashboard temp sensor was reading almost 3/4 and you can
feel the extra heat from the engine.

Then I tried removing the fan relay and put a jumper wire in its place
-- suddenly the fan started right up! I confirmed that the relay is
just not being switched on (checked voltage on the relay connector at
the fuse/relay box).

Then I put everything back together and put the relay back in place.
Bam! Suddenly the fan started up with the engine temp gauge reading
almost 3/4! I am thinking somehow it fixed itself or the connectors
were corroded in the relay connector, or something.

Stopped and started the car, back to nothing. The car's still hot and
the fan won't turn on again. Argh! But when I tried bridging the
relay again, it started right up.

So... I'm pretty sure it's NOT the fuse / relay / fan motor / wiring
from fusebox to fan / temp-sensor. It might be loose wiring or
corrosion in/under the fuse box but I'm not sure about taking it apart
- how hard is that? Or it might be the computer. Maybe it's sometimes
supposed to get up over 3/4 before switching the fan on -- but I don't
remember it getting that hot when I used to drive it! Or maybe
something else.

Oh, one other thing I almost forgot! There is a little one-wire sensor
(?) in the front of the engine, near the cooling fan. It goes out a
few inches from the engine and connects with a white plastic
barrel-style connector a bit smaller than my finger, to a wire which
goes (somewhere) routed next to the fan power wires. That white
connector thing used to be attached to a clip on the cooling fan cowl.
That clip melted or broke in the past, and the plastic connector sat
right next to the manifold on the engine and got partially melted. I
disconnected and reconnected it and it still seems to make a
connection. Perhaps unlikely this is the cause, but as the only thing
obviously damaged/out of place under the hood, I thought it was worth
mentioning.

I'm tempted to just put a manual fan switch in Mom's dashboard and be
done with it. But I'd much rather fix it right if I can figure out
how, and I wanted to get a second opinion. Thoughts?

-=Ivan