Furnace fan operates only when heating is on

I don’t have AC unit at my house, and summer heat is really bad when I sleep at night in my bedroom upstairs. Since I don’t have AC, I think manually setting the fan "on" while I sleep will at least circulate hot air in the upstairs and cold air in the basement to cool off overall temperature of the house.

My problem is, my furnace blower fan only runs while heating is on. Whether I set the fan switch to "on" or "auto" on my thermostat, it only runs while furnace is activated as if that fan switch doesn’t exists.

From thermostat (battery powered) to furnace control board, W – W, RH – R, G – G wiring is connected. On either side of the wiring, fan turns on when I connect jumper cable between R and G, but I don’t want to leave it like that. It seems too hacky and I would like to control when to turn on/off the fan.

I’m not sure how to troubleshoot the problem at this point, any guidance would be appreciated.

Thermostat brand is White-Rodgers and furnace is Carrier 58GFA086-14

Thanks