The reason for these problems is NOT Thunderbird or other apps.
I recently spent hours reading through these posts and their suggestions all of which did not work for me. There are many, perhaps hundreds, of posts complaining about folks being unable to login to or me.com using mostly third party apps like Microsoft Outlook, Mozilla Thunderbird, or other mail, contacts, and calendar services not provided by Apple. Here is a detailed response for resolving all these issues with Apple email accounts with non Apple apps or email clients.