I store my Thunderbird data files (meaning the actual e-mails/mailbox folders) in a separate directory on my C drive.

Kindly elaborate on the above. Do you mean you changed the Message Storage > Local Directory path for each account on the old computer?

Correct. The message storage location had been changed. I don't think that was the problem. I was actually able to complete the migration by using the Add-On "Import Export Tools NG". First I imported my old profile and then I imported the "folder with subfolders with structure" from the old computer and that seems to have worked. For some reason, some of the e-mails from earlier that day did not make it through the transfer (even though I closed the old program before doing any of the transfers) but good enough.

I am concerned, however, for dealing with this in the future. It does not seem enough to just back up the profile file and the e-mails because I cannot import them without that Add On Tool. So if Thunderbird stops working on this new computer (for whatever reason) I won't be able to export the files.

You can always transfer mail between profiles or computers by manually copying the mbox files - the large ones with no extension, named after folders, in Mail/<popserver>, ImapMail/<imapserver> and Mail/Local Folders in the source computer - to Mail/Local Folders in the target computer. That will work even if Mail and ImapMail are not located in the usual place as subfolders of the profile, as long as TB is pointed to the Local Directory in Server Settings.

Thanks. That's what I did. I copied everything over, into the the "roaming" directory of the new profile path on the new computer, but when I opened TB, it just kept asking me to set up a new e-mail, and did not display any of my accounts or mail. So something wasn't connecting. Either I needed to edit the profiles.ini file or something like that because it wasn't reading any of the prior data until I used that Add-On.