Are escape doors really activated because of the generators, or is the entity that activates the doors?
I'd say that the Entity allows the repaired generators to activate the escape doors. It does control the trials after all. So the answer to your question is yes. 😂
I'd say both. The Entity has developed these trials and intrinsically linked the gates to the generators. However, in the early tutorial there was a quote in the killer tutorial where the Entity whispers say: "Don't let the light in!", which suggested some urgency. The light does seem to affect the Entity as it's blinded by flashlights (can't see through the eues of the killer).
The stickler comes with the hatch. If one survivor is left and the hatch closes, the gates can be activated without the gens being completed. Yet, the hatch was not Entity-made, but rather created by Vigo as an alternate escape. This seems to bigger up the system, which is why I'm not wholly believing the idea the Entity has complete control.
Seeing as the generators are not even physically connected to anything besides their own attached lights, yet somehow managed to power both exit gates and lightbulbs/signs on certain maps, its safe to assume this is the Entity doing some dream logic stuff.