I was able to bypass the brake booster altogether and now the brakes work fine. I plan to replace the entire vacuum booster unit with new. I removed the white plastic cap and the white plastic disc and the rubber diaphragm, and could not find anything that appeared to be sticking or broken. Are there more moving parts under the black metal disc that could be sticking or broken?
I suspect what you have in your hand is little more than a breather. They get knocked off and clicked back together with no apparent effect.