when I first saw the picture, the work clean came to my mind first as well
Use self leveling calk right over the top of the old. Not sure why, but no one says to clean off the old. I assume it seaps into the cracks of the old and creates a new seal. Maybe it's just a waste of time or maybe it's just impossible to remove the old. Just relaying what I've heard. As a side note, Use LAP sealant to reseal around doors and windows.
Some people will apply eternabond tape over new seals to help them last longer. Guess you can never be too careful.