Completely agree with fummins it's like you have two separate issues.
The 406 connector contains the ground for the relay coil so disconnecting it powering off the relay suggests the relay is functioning?
If it were my car this is what I would do, after disconnecting the battery
For the load side and dismount the relay box under the hood and pop the bottom cover off, see if something chewed up the wires in there or if theres a printed circuit board in there that the relays and stuff plug into make sure it's not delaminated or signs of carbon tracing, black spots etc
If you want to look at the control side of things, ide disconnect c407 which looks to be the main plug in for the etacs ECU too the junction box and look at it. This actually seems like the most likely culprit right now except for the fact the lights stay on with the relay pulled
You could also disconnect c10 which is the data connector from the switch too the etacs ECU and see if the light goes off.
Fummins makes a good point that resetting the system makes a change, but I would personally start on the load side just because it's easier to understand and access.