There is even more simple solution which works better and doesn't have risk of getting other members having the same issue after one of them decides to update the household.
Your way will be a cat and mouse game with every member fighting for setting their household as "netflix household" and then other members will get the same issue.
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As per above snip I've taken directly from Netflix's docs, you can easily exploit this bypass they have given you officially by setting up a local VPN server on a small device like Raspberry Pi at the account owner's home (or someone from the group who can do it should have their home as Netflix household).
Then all you (other members) need to do is have your mobile which has Netflix connect once a month to the VPN of the member who's the account owner (actually, Netflix household) and play some movie/show for a while to trick netflix that your mobile is in the household. After that you can disconnect your VPN and connect to your home's wifi normally which also has your TVs connected on the same network and play some movie/show again on your mobile to trick netflix that you're travelling. By doing this TVs should start working again.
I wonder if this can be patched or not since it'll be hard to as they can't enforce this on mobiles due to their nature of being able to carry with you. The app can be made to detect VPN connections but that will trigger an entire other bunch of users who use VPN to bypass geo-restrictions.