Above senior you are allowed and expected to go "heads down" (by yourself or a small group) and knock out solutions to technical problems. However you are also expected to identify cross-cutting problems, rally the right people, coherently frame your arguments, etc to move the "implementing solutions to technical problems" needle (even if you don't directly write a lot of that code).
Management also get's the "fun" things like "let's talk salary!" or "you need to improve along X, Y, Z axis!" :P
Just my two cents :)
What you said about those cross-cutting problems and all that is spot on and that is basically what I see sr. engineers and architects. However, how do we or should we even further delineate career progression and responsibilities? And what does that look like? What is beyond those things?
But I've never been in the position to either be promoted or do the promoting to that level :)
Senior S/W Engineer -> Tech Lead/Specialist -> Architect -> Senior Architect -> CTO/VP-Technology