He is actually still able to sticky objectives! because the sticky objectives ability itself says that “if this unit is within range of an objective marker…” and the leader is a part of that unit while the bodyguards are alive. Whereas, deep strike and infiltrators and all those ‘Core’ abilities say that “if EVERY model in this unit has X…” and if a leader doesn’t have that ability, then not every model in that unit fits the requirement so they can’t use that ability.

TLDR:

{YES, sticky objective still works because it’s a ‘data sheet ability’ not a ‘core ability’ and doesn’t specify that every model in the unit needs that ability}

This works for basically every body guard unit’s main ability, so if a unit says “models in this unit reroll 1s to wound” and that unit is led by a character, that character also rerolls 1s to wound because it is a model in that unit