Debugging an old Thermostat: What component is this? (3 pins SMD IC)
If the relay clicks but doesn't deliver power to the heater, then the contacts of the relay are either burned out or there's a broken wire between the heater and the relay - or maybe the heater is broken.

You can't really expect to measure anything reasonable when the parts are still on the board.  The measurements of those SOT-3 parts are really questionable.  They connect to so many other things that there's no way for you to tell what you are really measuring.

Check the relay.  A relay that clicks but doesn't switch is probably broken.  If it is OK then check the wiring to the heater.