If you look under running the game you have references to failing by 5 for social checks and for traps. It's also shown in saves too. Sections that get into the topic specifically are titled Degrees of Failure and Criticsl Success and Failure iircIt looks like you are right, I can’t find much about degrees of success in the PHB or DMG. But a game without them would completely break and become unplayable, unless you were intentionally running some kind of dark comedy.
If he does any flips at all that is degrees of success, whether he botches the last one or not.
Thry are under the larger section called Resolution snd Consequences in the chapter Running the Game and one section I will quote hopefully ok is this
"As a DM, you have a variety of flourishes and approaches you can take when adjudicating success and failure to make things a little less black-and-white."
Then that section goes on to go into some cases and examples of mechanics and approaches.
They dont actually label a section degrees of success - maybe that throws folks off.