On her computer, why not just use Thunderbird on it? Or even outlook, or whatever she likes. She just needs to pick the software.
On her phone, or even yours, why the stuff with accessing Thunderbird through vnc. Just add the server to whatever mail app on your phones?
If you want a web based thing, roundcube or sogo. But Thunderbird is gonna suck the way you are trying to use it.
Restrict the number of accounts that can join that family group. And/or remove the ability to share the library from the main account for repeated offenses.
Or require multiple family members accounts to have to cheat before the owner account is banned.