Who says you have to be limited to a 2500 watt generator? And even if you are, the work to install the breaker and interlock is the same, or less than installing a transfer switch at the furnace. And with the interlock at the panel, you can power anything you want in the house, within the 2500 watt limit, or whatever the generator capability is.
You can power up the commonly used outlets, so you can plug in your phone, watch TV. You can power up the bathrooms, so you can see to pee. You can power up all the light circuits in the whole house, and just turn the lights on that you need, when you're in that area. Power up the fridge, so your beer stays cold. Power up the garage doors, so you can open them. I'd say that beats just powering the furnace and screwing around with extension cords. One extension cord to an inlet at the panel, power anything you choose.