Even if you accepted a spotlight, the more problematic issue is that rimward (southern) areas would experience far less time in the light, and the intensity would be less as well.
Imagine flat earth with diameter D, and a spotlight traveling along the equator. In order for a location on the equator to have a 12 hour day and 12 hour night, the spotlight would have to be larger than 1/2D. If this were the case, locations near the north pole would always be in the spotlight and all locations south of the equator would have days less than 12 hours.
This is basic geometry.
Observations show that during December days are longer in the south than in the north, a physical impossibility if the sun moved in a circular pattern above a flat earth.