Hard to follow descriptions

also, those brakes are likely crap and perfection is probably asking too much of them.

I'm guessing that the spring is assyetrically balanced, hence one side of the brake prefers to move more than the other. Combined with a lack of cable tension; this could produce the symptoms you describe of: only one side moving.

Try this:
unbolt the cable from the brakes
hold the brake closed against the rim (a 3rd hand helps here)
pull the cable tight
rebolt the cable onto the brakes

recenter the brake so neither side drags the rim when not in use
likely wont be able to make both sides contact at the same time; so just focus on neither one dragging