Hello,
Thanks for everybody's replies.
It is nice to know that it would have worked the other way around: i.e. use Gmail's contact info with either another identity, or simply a copy of its messages to the backup server.
Yet, I will likely continue to use my current configuration as long as Google still allows "less secure applications". It seems to me easier for 3 reasons:
– No need to reconfigure Thunderbird and all filters.
– Messages are stored out of Google, where I ultimately want them to be.
– If I have my personal Gmail account and the corporate Gmail one too, my telephone constantly asks me which account I want to use – for mail, text messages, photos, maps, agenda events, etc. It works, but it's less fun.