Could just bypass with esp32 and relay. I think you just need to close a 24v AC switch/relay unless you have a newer/more expensive communicating system (although then I don't think you'd have an Ecobee)

i have an ecobee and it was pretty simple to hit fan on even when system is off. this may be a limitation of your hvac system being exposed awkwardly through to ecobee. some systems have a "fan only" wire, which mine does. but my guest house is a much older system, and it is like yours.