It does, but your client creates the message and deposits it into the server outgoing mailbox. The content of that mail, text, images, hidden text, HTML etc, is controlled by your client.
When you use webmail, you are logging in directly on the mail server. When you create the mail and click send the webmail deposits it into the server outgoing mailbox.
However, with the webmail, the content, text, images, hidden text HTML, etc, is governed by the code in the website which generates the mail. You have absolutely no control over it. So if a mail created by BT webmail fails to be sent because it contains unacceptable content, when the only user content is some plain text, then it is the BT site code which is filling it up with filter breaking junk. Therefore entirely BT's fault.
You can see the mechanics of connecting to and sending a mail, by hand direct on the server,
here. Note I used an article including the OpenSSL process because there are no internet connected servers without SSL. Not should there be any internal one's either.