My '17 EL just started doing the same thing so I came here to look. Blower won't come on, kick under the glove box and it starts up, etc. I thought it might be the motor since it is really loud on the higher speeds (separate problem - anyone else have this?).
I got under there to look at it with a thermal camera and you can see the issue. The connector and terminals don't look melted at all, but the terminal has pretty high resistance and must barely be making contact.
This is after about 2 minutes of running at 75% speed:
You can see the 2nd wire in (yellow w/green I think) is pretty hot. I pushed the connector on really hard and zip-tied the wires up so there's pressure on it and hasn't died since, but figure it's only a matter of time before it fails so I'm ordering the replacement connector.
Just curious -- the harness is plugged into what appears to be the "resistor pack", but it behaves more like an electronic speed control w/4 wires. I noticed it would over-speed the fan when I jiggled the connector like a digital ESC would. Did any of you have to replace that part as well as the connector when it was melted/damaged?