Maybe the prototype θύρα would need more description - a close-able opening in a house (smaller structure), more of a hole with four sides. I'm not sure if the prototype would focus on the hole or the closing. When we say "door" we definitely think of the bit that closes the hole.
In contrast, the prototype of πύλη might be a close-able opening in a large structure like a wall, not necessarily bordered on the top.
θυρα and πυλη occur tied together --> θύρα(ι) τῆς πύλης.