Yes, I have physical control of the computer and no one else has access.
You seem to be saying that when I first entered the account into T-bird, that was only time a password was necessary. Is that right?
So you are sayingo this is normal behavior for T-bird, right?
By default this would be the normal behavior. Virtually no one wants to be prompted for the password for their e-mail account each and every time the e-mail client they're using does a send/receive cycle. They just want it to do a send and receive.
At one time, when you were setting up an account, there was a way to specify that it was not to save the password and you would be prompted each and every time the client needed to connect to the servers. I never do this and no one I know does this as it's so darned inconvenient and, if you have physical control of your device(s), completely unnecessary.
If that were true, why is T-bird asking me for the email account password at the same time it makes the account accessible without the password?
You do not seem to be making the distinction between "your account" and "your downloaded messages" from the account that have already been snagged by the e-mail client on a prior send/receive cycle. If it's asking you for the password access to the account, on the server, is not possible without it. Anything that has already been downloaded and is inside the e-mail client's database of messages is.
Exactly which messages will be available depends on exactly you have configured either your POP or IMAP access to your e-mail server for a given account.
It is not the responsibility of an e-mail client to keep the actual user of same locked out of their e-mail. Thunderbird does have a master password capability, but in reality if you're using a Windows machine that has the capability of establishing separate user accounts that is the mechanism that should be used to keep someone from accessing not only Thunderbird, but anything else related to your account.