Glad you got it running.
For future readers...
A pump is never shot if the pump is running. It can be fixed. These pumps consist of a motor and a head. The head is typically a diaphram made of rubbery flaps. The flaps might be bad, torn, degraded or clogged with debris, but can be replaced or cleaned. A little spec of plastic bit is enough to keep the flap from closing fully and if it is full of air, not water, then it will not be able to pull nearly as strong as it could pull on water. That one spec of plastic debris is enough to prevent priming in a dry pump.
By filling the tank full you are putting more pressure on the water to be pushed into the pump, so that helps overcome the issue, whatever issue the pump was having to pull.
As mentioned, the pumps are not at all cooled by the water, so can run dry with no issues, the motor does run dry even when pumping water so-to-speak because the water never cools the electric pump anyway. The pump head has water, the motor does not.
Actually a tiny spec will likely just pass on through to your faucet, but it is the debris that is more fish hook shaped that will get caught in the pump head. A filter or just a wire screen thing before the pump will prevent this.