Questions: should I connect this? Also, if needed how was it working before without this connected?
The basic thermostat interface to a furnace is very simple.
Red supplies power, if that is connected to white, the thermostat is saying “turn on the heat”. If that is connected to green, the thermostat is saying “run the fan”.
If your thermostat runs on batteries, (or is an old mercury-switch one that doesn’t need batteries) then there isn’t a need for “common” to be connected.
If your thermostat uses power supplied by the furnace, it (probably) needs common connected. The power and common wires will provide the energy to the thermostat so that it can run it’s electronics and connect/disconnect the red-to-white and red-to-green connection.
If it were me - I’d connect common (black or blue) in the furnace and at the thermostat, and stop buying batteries for the thermostat.
But I’d also buy a “smart” thermostat so that I can turn up/down the thermostat from my phone. That way if I’m away for a week I can turn the heat off/down, then turn it back on/up as I’m driving home.
(And I do own an ecobee thermostat so I can do that as well as set schedules, etc.)