Another vote for the levers as the primary cause. Easy to measure the cable pull to compare new vs. old - that way you can confirm or rule out. I have 2 bikes with CP brakes (DiaCompe and Weinmann) and they work adequately. Both have narrow tubular rims so width is not an issue.

Cleaning the rims may help, but will give only marginal improvement (unless you did something like a friend of mine did - wax the rims to stop brake squeal - not a good idea).