Can this make sense, or is this just some weird coincidence?
Yes. The easiest(hopefuly) thing to do is verify coolant level as it drops enough if low when turning on heater is on causing your guage to rise.
Another case is potentially a clogging radiator with poor pass thru. Basically all the coolant passes the radiator thru the heater core which does not offer much in cool down capacity compared to car radiator.
Good luck, had both problems.
Also when flushing and refilling coolant, have the heater set on Hot. That opens the valve to the heater core all the way. Let the car idle for quite a few minutes with the radiator cap off, heater on High, to help expel trapped air.