Connector that keeps the Switch fan connected broke off

The little clip is just to keep the cable squished into the connector pins, and to keep the cable from coming loose, if the pin damage is just on the top part of the pin, I don't know that that's a huge deal and would probably just be cosmetic . Seems logical that this is related if the clip is broken off and the fan isn't spinning. But all the contacts for the fan cable are on the bottom side of the fan cable and since the cable is plugged in in your picture, it's hard to tell if there is any damage to the pins in the connector where they make contact with the cable. I would unplug that cable and inspect the connector for any other damage.

If no, it may just be that the cable sitting flat in the connector is not enough to make a strong electrical contact. See if you can use a plastic spudger or something flat and not conductive to press the cable into the bottom of the connector and see if you get fan spin that way. Otherwise, I would check for continuity with a multimeter to make sure the connector is seated firmly on the board. If you do need to replace the connector, yes you would need soldering skills for sure. If you get fan spin with pressure, I would see if you can just get a new connector and harvest the clip from that and install it on the existing connector.