They don't. As per the PHB page 194 this is what is said about 1 and 20:
ROLLING I OR 20
Sometimes fate blesses ar curses a combatant, causing
the novice to hit and the veteran to miss.
If the d20 rol1for an attack is a 20, the attack hits
regardless of any modifiers ar the target's AC. In
addition, the attack is a criticai hit, as explained later
in this chapter.
If the d20 roll for an attack is a I, the attack misses
regardless of any modifiers ar the target's AC.
That's for attack rolls. no equivalent can be found in the pages covering ability checks or saving throws.
This is correct. As I mentioned in the OP, attack rolls and the saves vs death are the only rolls in 5e that are guaranteed to fail on 1 or succeed on 20. Literally every other save or ability check just compare the result numbers. So that DC 5 lock? Once you have a +4 modifier, it will be a 100% success rate. On the same ticket, that dc 20 lock is impossible if a character has even a -1 penalty.
That is the way it should be, though. So frustrating to have +13 slight of hand and still fail a DC 5 lock, and so stupid to be able to save scrum a dc 20 check with a -3 mod until the 20 gets rolled. And before you think those chances are rare, remember that rolling a 1 or 20 is basically a 10% chance with disadvantage/advantage, respectively.
