That's a quick, easy test that would produce some useful info. If it responds then it rules out the wiring to the thermostat and the thermostat fan switch.
I've also see older furnaces where once the blower turned off, a minute later it would come back on again, because the low temp turn off was set low and so was the fan turn on point. So, the residual heat would make it start up a second time.
My old furnace here, if I had the temp set way back to like 45 while away on a trip, when I came back and started it up, the reverse would happen. Upon startup, after a couple mins the blower would come on, but after about 30 secs, it would shut off, because the temp had dropped enough to cycle it off. A minute later it would be back on. It would do that a few times until it staid on permanently.