I imagined constructing multisig addresses of a set of (potentially completely independent) BIP32 chains, using equal indices in each, in lockstep.

In my opinion, BIP32 is mostly about generating sequences of keys, which can either be used directly as pay-to-pubkeyhash addresses, or combined using multisig into a P2SH address. Isn't that both more simple than trying to put BIP32 on top of multisig, and no reason for having the chaincodes be independent?