Cold Basement - Old Gas Fireplace

A stove will radiate heat and that's how it warms. Convection (i.e., fan) will speed the process. That said, some stoves send less heat up the chimney than others.

Maybe get it serviced/tuned. Flames should be mostly blue with a bit of yellow (blue is more efficient - but flame is made yellow - which indicates incomplete combustion - for ambiance.) A new furnace with installation may end up being factors more expensive than fixing.

As for basement being cold, either humidity is excessive (possible) or the heat is leaking out somewhere.

Change the thermocouple and add a fan. Both are simple to do.