the bridge is to be removed if you have a second heat source from an ac fan coil . like a boiler and radiators. this way the thermostat seperates the two control circuits. R1/W1 being an independent dry contact - a discreet switch that closes and energizes the boiler and the R2/Y/G taking care of the fan and cooling. If everything is at the same appliance, your furnace , the jumper stays.
new vs old.. new will definitely allow independent fan control an older furnace may simply sequence the fan on after lighting off the burner. New will likely have 2 stages of heat vs single stage on an older system. The idea being the first stage provides lower blower speed and less gas consumption - quieter and more efficient. If it fails to reach the setpoint after a specified duration the second stage is initiated. The very latest equipment modulates across a range of capacity based on demand… = way more service calls :cry: nothing new is reliable.
a new stat will control an old furnace fine over the 2 wire connection - you'll still realize the savings gained by automatically reducing the temperature when no one is home. That is the whole point of the Nest et all. If you take 10 minutes to program a $40 home depot stat it'll do the same - in fact as long as it uses batteries it's the perfect solution for a 2 wire heat system - no new wire required.