Rubber tip on the brake rod

NOT needed for power brakes. The pushrod is captured inside the booster linkage.
As far as manual, in theory an inferior "retainer" grommet could fail and allow the pushrod to pull rearward and slip out of the hole.
Slipping out of the hole is rarely a good thing. I've never had that happen with a brake system even though I've parted out several old cars and found the grommet either hard as a stone, worn out or missing.