I had an issue like this once. In my case there was an issue with the mechanism that read the temperature in the thermostat, so a new thermostat fixed the problem. I have no idea how you test this without buying/installing a new thermostat.
I had the same experience as Clawdio. A new smart thermostat (which was almost free, since it was heavily subsidized by local energy efficiency programs) fixed the problem immediately (and considerably improved home comfort by circulating the air more sensibly).
If you don’t want to do that, maybe you could lookup the pin/wire diagrams for your HVAC unit and figure out which wires you’ll need to connect in order to turn the fan on? They’re all different, though some configurations are more common than others.
Once you do, then you can look at the relevant wire terminals and soldering between those wires in the thermostat base. Maybe there’s some corrosion or something else there causing a permanent short, or a failed/degraded switch mechanism like in Clawdio’s case.
In any case, a new thermostat is probably the simplest (and possibly cheapest, due to numerous subsidies) fix.
We had the same thing happen right after our HVAC was serviced in the spring. I called the company back, and they did something at the unit, not the thermostat. All I know is it works now and it didn’t cost anything since we have a service contract. I know that’s not much help, but I find it interesting that this seems to be a non-rare failure event.