Jeb! Bush really seemed to think he could pull it off.
He jumped into the Presidential campaign with a logo that left off the "Bush" name and added a "!," and distanced himself from his brother by telling voters he was "his own man."
When that failed, after the campaign math equivalent of about 10 minutes, the old habit of relying on the old family business to get ahead for nothing kicked in. The problem is that many voters aren't so stupid as to forgive someone who very nearly killed the country and twisted the knife on his way out the White House door, especially when the wound is still raw. Handing the country over to the next bad apple on the family tree is a tough sell. Just telling folks you're the "smart" apple and trying to shine yourself up (and failing miserably) won't change the fact that you're still just a piece of fruit.
And so it ends for Jeb!, who finds out the hard way that his brother not only didn't keep the country safe, but he couldn't do much to save Jeb! either, and Jeb! ignored the advice of the one person who actually could have when she said "There are other people out there that are very qualified, and we’ve had enough Bushes."
Jeb! should have listened to his mother.