is it better to run a scooter motor toward hot or cold?

If the needle budges off the peg, then your scooter is heating up just fine. Cooler is better, so long as it's getting hot enough to warm up the oil. Within reasonable limits, the cooler it runs the longer it will last. It's probably designed to run towards the cooler side so as to have some reserve when it gets bogged down in traffic on a really hot day.

Always give year, make, engine size and model # when asking questions. We need all of that info to give an informed answer.

A stuck closed thermostat would cause the engine to overheat.

It opens when hot, to allow coolant from the cold radiator to circulate through the engine.

It's mechanically preferable for the engine to run cooler, rather than hot - less ware on the engine.

There will be better performance and gas mileage at the proper operating temperature.

If the engine runs fine when it's cold - then it's running fine.

Depending on the model of your scoot, cheap models use inferior parts, have poor quality control and the manufacturers don't care how accurate gauges and sensors work, to save on costs.

Sounds like the thermostat has failed in the open position, or is missing altogether. Change it out. Remember you may get an air bubble when you do, most of them have a place on the housing that you can open to let air out.

I would say hot but not too hot

this is why i like air cooled engines. the thermostat sounds like its stuck