"But Thanos and Strange have no relationship to speak of, so there must be some kind of handoff going on."
Just to make a point about the above statement... an IC doesn't have to directly be engaged with an MC in order for there to be impacted. For an example, there's Pieces of April. Chris discusses this in a podcast (sorry I don't have a link). The mother and April, MC, don't actually interact with one another until the very end of the story.
So, it's possible. Strange does alter his view regarding the time stone, unwilling to sacrifice to willing to sacrifice. But for the "greater good."
Reminds me of Vision saving Cap, taking on Cap's view. But that seems more thematic. Anyway, I was just scanning this thread and came upon your point about Thanos and Stange and wanted to throw that thought in.
It's important what the audience synthesizes, ultimately, that makes "impact' relevant.