Allay is able to go through double doors, but can not go through single doors.
To reproduce:
1. Make walls and ceiling
2. Start with double door. Make allay follow you out the door
3. It usually works through double doors. Now try a single door.
4. It will not go through.
In fact, he can go through 1 door, but he tries to do it by leaning against the door. Therefore, most likely this is a problem of the intelligence of the mob.