this may clarify the thermostat switch position functions. the on/auto switch is basically a fan override allowing you to force airflow at any given time, regardless of the system switch position, even if it is the 'off' position. this option has always been available on the modern low voltage thermostat (since the 50's). it can be useful to use the optional 'on' position during the cooling season if you have remote areas and want to keep the air moving, thereby providing a mix of all the air in the conditioned space. this also provides an average temperature as it is pulled past the thermostat and into the return grill. you could also use the option during a dry spring or fall day if you want to crack windows and circulate the cool air through the house. in the case of the auto/on switch, 'auto' simply means that the cooling fan speed will be energized simultaneously with the call for compressor operation. two simulatneous signals energize the contactor outside and the fan relay inside, then de-energize together. placing the switch in the 'on' position forces the fan relay to stay energized when the compressor signal is de-energized.
the lower speed fan function does not initiate from the thermostat at all in heat mode. that signal is part of a timed sequence programmed into the circuit board and used in conjuction with sensors and safety switches to ensure a safety shutdown in the event of the accumulation of too much heat in the furnace. however, it is a system made up of electro-mechancialdevices, subject to malfunction from time to time. if the heat comes on and the fan does not, the problem can only be in the motor or the board, or possibly a wire. limit switches. strategically placed for optimal sensing, can shut down low or high voltage circuits. a limit may shut down the whole operation, but it should be noted that a limit does not prevent a blower if the heat source is energized. on the other hand, if air doesn't move, the heat source power supply will be interrupted by a temperature sensing device, commonly known as a limit switch.
if you turn on the fan switch during heat, the air moves faster, creating a draft. when the heat source de-energizes, the draft will become significant.
by the way, flame sensor and induced draft blower air flow sensor hose would have no bearing on the problem as you described it. perhaps that was a different service issue. beware of untrained field technicians.