I seriously doubt what you are saying is correct, but I am not saying it is delivered.

There is exactly no guarantee of delivery in the email specification, just at paper letters have no guarantee of delivery. This loop hole is exploited by providers in their fight against spam, as they d not have to deliver they can filter anything as spam and not be accountable. My guess is your mail is being classified as spam.

I suggest you ask your mail provider is they are sending out mail submitted to their SMTP server (don't let them distract you with web mail. It is not the same. They provide the SMTP server at it is up to them to support it as well) My guess is they are and the mail is being dropped by the recipients server. If your own SMTP server was refusing to deliver your mail you would be getting an error message at the time of submission. After all that is why the sending box actually takes so long, so that any error in submitting comes back to you at the time of sending as a popup. I therefore conclude in the absence of an error information from you that the problem lies further down the deliver path.

BTW this sort of problem is quite common with small business web sites and their shared host plans, you get labeled by the mail actions of the others sharing the server.

I have 2 laptops, side by side. Both are using the same updated version of Thunderbird. Both, also have my same 3 email accounts on them with identical settings. It is just one of my laptops having this issue, and only with one of my email accounts. I can send from my other 2 email accounts with no problem. I can send from my other laptop from all 3 email accounts with no problem. I can send from my phone with no problem. I can send from webmail with no problem. I really don't think this is mail provider issue, as all the other accounts and all my other devices work.

And I do not think it is a Thunderbird issue as all your other accounts work fine.

So again I will ask you to contact your provider and ask them what they are doing with mail that is submitted.

You can log the connection to the mail server if you like to prove the mail is sent to the server. https://wiki.mozilla.org/MailNews:Logging