Auto works but Thermostat fan "on" does not

Heating, Ventilation, and Air Conditioning (HVAC)
1

i have a trane xe80 furnace about 13 years old.
air conditioner installed about 8 years ago.
programmable thermostat - also 13 years old
Everything works good
EXCEPT.
when i turn the thermostat to ‘ON’ from auto nothing happens. (i.e. the fan does not run)

i want to leave the fan on all the time to circulate the air.

at one time the fan “on” worked, and it also worked after i had the AC installed as well.

perhaps the thermostat is shot.

any pointers on testing this. i have some electrical skills and understand most principles and know how to use a volt meter and that it is not a good idea to stick a knife into an electrical socket.

2

see attached picture
OK, so i looked at the wiring and think i know what is likely wrong.
4 wires go to the thermostat.
Red to R
Green to G
Purple to Y
Black to W

with thermostat on auto i have 24 Vac from red on thermostat to each of the 3 wires, turn thermostat from Auto to on and voltages to green drop to near zero.

checked out furnace wiring.
there are 3 sets of wires as follows
Wire 1 - to thermostat
Red to R, Black to W, Purple to Y, Green - taped off (not used).

Wire 2 - to air conditioner
    Black to Y, Red to C

Wire 3 - to exhaust fan
     Black to R, Red to G

So how this is set up from my understanding is that the fan kicks in when we turn on the exhaust fan.

However, i want to be able to turn on the blower fan without turning on the exhaust fan. in winter i am loosing heat by doing this, and in summer i am loosing cool air.

my question, can i hook up the green wire (from the thermostat) to the circuit board (G) so i can use the thermostat to run the blower without turning on the exhaust fan.
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