Natural heating and cooling

For the last 20 years, I've had two or three months when I just use a whole-house fan for heating and cooling, taking advantage of cool nights and warm days as available, and relying on insulation to keep comfortable the rest of the time.  Using a timer on the fan is often adequate, but to maximize the benefits, I sometimes lose a lot of sleep.  I recently learned that there is a name for the hardware I want.  A differential thermostat can compare the inside and outside temperatures, and flip a switch when they match.  I think that if this were offered as a simple plug-through gizmo with one remote sensor, a lot of us would find it quite handy.  Has anyone seen anything close?

That does seem like something missing from the marketplace. I thought for a minute that this: https://www.amazon.com/Inkbird-ITC-608T-Temperature-Controller-Thermostat/dp/B07MB16C3K/ would do that, but it's not smart enough to have a differential mode.

Hubby built something similar but different decades ago, so he says the concept is out there and available.

He spotted this one on Amazon: https://www.amazon.ca/Differential-Temperature-Controller-Thermostat-Greenhouse/dp/B0C387DPZK?
which he thinks will do the job.  Of course the devil is in the details. Unfortunately the Amazon listing is very limited in the technical details provided but it is clear that the device will work as a differential thermostat which is what you need.  One concern seemed to be mostly that fan motors draw more juice than many of the controllers he spotted are designed for but this can be handled with an appropriate relay between the differential thermostat and the fan motor.

Much of this sort of thing has leaned far into using microcontrollers to do the controlling, rather than simple switches. He drew a quick diagram of the old school approach using a couple of LM335 temperature sensors, an LM339 voltage comparator and a relay , but you would still have to be able to source something to house the parts and some sort of power supply (wall wart?) that fit the requirements.

He did strongly agree that what you're wanting is out there for things like solar water heating units etc. The trouble will be matching what's available with what your equipment requires safely. An electrical fire while you're asleep is a bad idea!