Why don’t we just call ourselves programmers and save this semi-regular debate!

Because that diminishes the work people do. A programmer takes logic and encodes it for a machine to execute. Being an engineer suggests solving problems and defining logic.

The engineer title is apt in my opinion, because if you look at construction as a parallel, the architect designs the shape of the building, engineers determine how to build it so it doesn't collapse, and builders actually make it real. Programming is like digital building, the architecture and implementation details are both separate.

Because I haven't been allowed to program for 6 months. Now I'm just an AI manager.

Centuries of accumulated respect for the word "engineer", implying a specialization that can't be simply replaced with other skill sets. That is: prestige and therefore economic incentives.