For one of them I have unchecked all the server settings: "Check for new messages at startup" "Check for new messages every ..." "Allow immediate notifications..."

This is an IMAP account, right? Unchecking those settings wold be enough for a POP account, because a POP client only needs to connect to the server to check for new messages, but not for IMAP, because the nature of IMAP is that everything is stored on the server and hence the mail client needs to connect to the server to remain in sync, not just to check for new mail, so every time you tell Thunderbird to show the contents of a folder, you're telling it to connect to the server for synchronisation reasons. Using an IMAP account, meant to store mail on the server, for local archiving purposes is fundamentally wrong. You're trying to use the account for something it's not meant for.

You need to decide first whether you want to keep your mail stored locally on your computer, remotely on some server, or both, and then use the right tool for that. For mail to be stored remotely on a server, you need an active IMAP account. For mail you want to keep stored locally on your computer using Thunderbird, you may use Local Folders, at the bottom of the list of accounts in the folder pane. You may create folders there to store and organise your mail locally on your computer however you wish, independently of any mail account you may have.

As long as the account is there, Thunderbird will attempt to contact it. The only solution is to remove the account .