You have an application in your computer that will be sending mail at a specific time so of course the computer has to be on, connected to the internet and Thunderbird has to be running. Otherwise it will send the message when it is on and in a connected state.

Thank you for your answer. The "of course" part was not obvious to me. I was hoping that the application on my computer might set up some timed task on the mail server (a crontab task in Unix/Linux and hopefully something analogous on an Exchange server and on whatever iCloud and gmail use), which would then no longer require further action from the application on my computer. One can always dream though...And current functionality of the Send Later add on is still a godsend, even if I have to make sure my laptop is online at the necessary times. Thanks again!