The vibration of the pedal is probably the anti-lock system activating.
I was thinking that, but if that is the case- I would think they should not be. This has happened during breaking on fairly smooth, straight, dry roads usually at less than 35 miles an hour. I'm wondering if a bad or failing master cylinder would activate the ABS? (As stated, we have bled & power bled the brakes several times, and they are still marginal, even immediately after bleeding.)