There is a relay in the spare wheel well. One side of the thermostatic switch on the carb should have continuity to the chassis, the other side continuity to pin 86 on the relay.
Pins 85 and 30 on the relay should have permanent 12v on them. One side of the fan motor should have continuity to the chassis and the other wire continuity to pin 87 on the relay.
When the thermostatic switch closes it engages the relay switching 12v on pin 30 to pin 87 connected to the fan motor. As the 12v is permanent the fan should operate without the ignition.
If continuity between contacts is ok and shorting the wires of the thermo switch together doesn’t operate the fan, replace the relay.
Right now on the sensor on the carb I have the black wire with a purple line that goes right to chassis. And the other wire on the switch is grey with a red line. If I touch them together the fan comes on. But I have never heard the fan come on before. They’d why I thought something was wrong.
It might do, if one is fitted but not switching on (iirc this carb cooling fan wasn't fitted to all models of the X1/9, it depended on destination market). The purpose of the carb cooling fan was to try to prevent the fuel in the carb becoming too hot and either turning to vapour or expanding and flooding the engine, which could result in difficult hot starting.
The fan temperature sensor unit is available from some of the X1/9 parts suppliers.
But there can also be other reasons for a hot starting problem e.g. an ignition system problem. I'd suggest getting the carb fan (if fitted) back into working condition, then if no improvement when hot starting, to check out the ignition system when hot, before going near the carb. (it could be a faulty float valve or float level setting).