Sending directly to user's MDAs is considered bad form bordering on malware. It chokes the Internet and mail servers with myriad short connections delivering but a single SMTP message payload.
The system was built with the assumption that most originated email will pass from your MUA to a local MTA, which will aggregate emails into multi-message payloads to be delivered to intermediate MTAs that sort and forward ultmately to the remote destinations' MDAs for eventual pick up by addressee MUAs. What you are doing will work, but the major players in the SMTP mail network are doing all they can to discourage it. The primary reason is spam, but the excessive traffic resulting from direct MUA-to-MDA SMTP is also a factor.
Why fight it? Just send this email via your local MTA, generally provided by your ISP. Almost nobody allows anonymous relaying anymore, so you'd need a valid account of course. This also neatly resolves the issues you're having with heavily protected email providers like those you mentioned.