Many thanks to reader ajcg whose comment in my previous post was spot-on. Once I upgraded VPC 2007 to SP1, then the Virtual Machine Additions version that got installed into the guest (Win7) was 13.820 which was recent enough to work on Win7! Yay! More recent versions of VM Additions should work as well, but 13.820 seems to be the latest I can find externally.
Incidentally, once I searched on 13.820 I noticed a comment left by MikeH2005 that confirms the SP1 update of Virtual Server 2005 R2 includes the same 13.820 version (and is what's needed for Win7), so if you're running Virtual Server, then SP1 should fix you up as well.
Download links to get the SP1 updates: