I run my house with an HF 4375 generator, 240v. Runs my well pump, gas furnace, lights refrigerator. I can cook, watch TV. Just can't do it all at once.A 2kw will easily run his furnace blower. That was the only thing he was interested in running.
Yes, a bigger generator will run more of a load at the cost of consuming more gas.
Runs almost 16hrs on 4 gallons of gas.