The main error is actually the other one, though I'm not sure what the full error is. What's the full message provided by the log that references "450 4.2.1 Too many..."?
It's either going to indicate that there are too many messages in the connection or you're sending too many messages. The latter would relate to quotas and this isn't something that XF has any awareness of.
Unfortunately you've still cut off the specific line I need to see. I need the full error message itself. It's above the stack trace that you showed. It will include the "450 4.2.1 Too many..." message that was cut off in another of your screenshots.
Swift_TransportException: Email to ....@gmail.com failed: Expected response code 354 but got code "503", with message "503 5.5.4 Bad sequence of commands. " src/vendor/swiftmailer/swiftmailer/lib/classes/Swift/Transport/AbstractSmtpTransport.php:457
To clarify, there are 2 distinct error messages in the screenshot in your first message. They appear to be similar, but the specific error message in each is very different. You have only shown one of them in full detail. Each time, you've shown the one with this content:
Expected response code 354 but got code "503", with message "503 5.5.4 Bad sequence of commands. "
I need the full error message for the one that says "Expected response code 250 but got code "450", with message "450 4.2.1 Too many messages..." This error is the primary error -- the "bad sequence of commands" is almost certainly caused by the "too many messages" error.