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it's not centering properly
Make sure the back nut, stationary arm and centre bolt (red arrows) are locked together, which they should be as a result of the back nut acting as a jam nut against the thread in the stationary arm, and that the set screw (blue arrow) is properly set against the M7 thread on the centre bolt. If your cable is still pushing the brake off centre, it's usually due to a too short cable housing, which is particularly common at the rear.
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I've just taken apart my side pull rim brake caliper, cleaned it and lubed every moving part. Is it possible that the friction in all the parts is so low (as in: everything moves/rotates very easily), that the force of the brake cable pushes the caliper to one side, resulting in one brake pad hitting the rim when the lever isn't pulled (i.e., it's not centering properly)? When I pull the brake cable and pull the lever, the caliper indeed seems to center somewhat/better.
I've watched the Park Tools video on how to adjust your rim brake, but to no avail. I feel like I've played with all the nuts and bolts to adjust the tension/friction. Adjusting the small alignment screw also didn't work. Very weird problem.