Man, I really don't want to just start throwing parts at it…

You're saying the starter relay is a common failure? If so, I can swap that fairly easily. It's worth a try.

Doesn't a failing ignition switch usually have other symptoms? Other than the no start today, everything else seems normal.

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2002 X5 4.6is - Imola Red/Black (Sold)
2001 740i M Sport - Anthrazit Metallic/Black (Sold)
1998 M3/4/5 - Technoviolet/Dove (Sold)
2003 Z4 3.0i Sport - Sterlinggrau Metallic/Black (Sold)
2009 Mini JCW (The Wife's)
2015 i3 REx - Arravani Gray/Tera World (Daily Driver)