I don’t know all that much about balancing wheels. A quick internet search on wheel balancing leaves me no wiser. The way I see it, you put the wheel on its shaft and see if one part of it always rotates to the bottom. If so, one side is heavier than the other and weights need to be added to the opposite end to balance out the anomaly. The flywheel seems to have a heavy side but I tried to mount weights (sawn-up iron mending plates) temporarily using tape instead of screws and couldn’t get them in a position to ease the problem. The wheel rotates just fine when treadled so I am not worrying about this too much – I’ll come back to it later if anyone has any advice or suggestions.
I made the wheel round with respect to rotation about the axle using a gouge and make-shift tool rest. The gouge left quite a mess behind in the form of small ridges in the wheel and tear-out where the grain abruptly changes direction (Underhill was right to suggest carving the groove by hand rather than on the lathe…). I tried to use sandpaper to clean it up but it didn’t do all that much other than slow the wheel with all its friction. I ended up using a low-angle block plane on the spinning wheel and this worked very well.
The treadling action is smooth but the wheel does wobble from side to side quite a bit. I cut in the groove for the rope by keeping the gouge in a single central location. The groove weaves from side to side relative to the wheel but is stable relative to where the pulley will be. I hope this all works when the pulley is installed!