One thing I dislike about multi-sig (and please correct me if I'm wrong) is that you can't back it up in the same comfortable and easy manner as a single-sig, because master public keys aren't meant to be written down on paper as seed phrases. So, you need to store it electronically, which is also prone to fail overtime.
Yes you need to back up the xpubs, and yes that is a pain/error-prone to do by hand. I don't think that means you need to back up electronically, though. You
can still write them by hand, provided you take your time, use clear writing, and triple check everything (including checking that you can successfully recover each share before you fund the wallet). There is also less risk involved in printing out xpubs than there is in printing out seed phrases or private keys, for obvious reasons. Provided you have set up your multi-sig properly (e.g. using 3 different and completely separate devices for a 2-of-3 multi-sig), then printing out xpubs from each device presents minimal risk.
And of course, you do not need to store every other xpub with every seed phrase, but rather
n minus
m xpubs with each seed phrase (provided you pick the correct ones, of course). Doing so also brings a privacy benefit since an attacker with access to one back up cannot even view your wallet.