I have tig welded a lot of flywheels as a preventative measure. If you wait until the rivets are loose enough to "rattle" you have waited too long. It is next to impossible to get the flywheel shell and hub concentric before welding after the rivets get loose. If the rivets are still tight and you see rust colored streaks on the flywheel shell that are originating at the hub and projecting radially outward, it is time to weld the hub to the shell.
I make 3 welds 1" long 120 degrees apart. I have used filler rod on some, fused the hub to the flywheel without filler rod on others. I have had both types balanced and could not feel any difference in the engine vibration between using filler rod and not using filler rod. If you are a competent tig welder, the weight of the welds will be really close to each other. Each weld will be within 1 or 2 dabs of each other when using .035 or .045 filler rod. I have welded some flywheels that the rivets were rattling and they would vibrate a lot because they were not concentric.