Furnace can’t keep up when set to ON but keeps up fine on AUTO. Is this normal?
Running the fan reduces stratification, meeting that it makes the temperatures in the house even. The thermostat being centrally located in a house is often satisfied before the remainder of the house is able to warm up especially in an old house where the heating registers usually are centrally located what's return air located at outside walls. In a house with no insulation the exterior walls lose heat much faster than the heat can be replaced from the ambient air within the house. Basically it's an indication that you have a poor duct system which is not uncommon in an old house.
If you let me know the state, square footage not including the basement if it's below ground, the furnace model number or the size in btu I can tell you a little more about your system.
Edit
Although the above statements are very accurate I did interpret your question in reverse. I read it as, you were having a problem maintaining the temperature with the fan set to auto.
Having your issue with the fan set on would normally be a real head-scratcher. But in this case I would say you need to have a look at your Ecobee. It's a fantastic thermostat, I have one too and there are a lot of features available to tweak your system. If you have multiple sensors and and they are satisfied before the main thermostat the system may shut off prematurely.
This is a  screenshot of my ecobee

If the sensor upstairs reaches the temperature set point well ahead of the main thermostat I think ecobee uses some kind of average to determine when to shut it off.
To determine if this is the issue I would suggest turning off all the sensors except for the main thermostat run the fan set to on and see if anything changes.