I mean in that case these services are just sending the email on your behalf, so it's their IP address that is included. But if you're sending the email yourself then it will be your IP address.
That might have been a valid argument in 1999, but in 2019 it's difficult to send an email yourself from a consumer IP address and actually have anybody on a major provider receive it. Major email infrastructure doesn't support those users any more; they're mostly irrelevant to the discussion.
I don't think he meant sending the mail yourself to the destination domain directly (which would indeed be blocked due to plenty of reasons), I think he meant connecting to your email provider's SMTP (over the 587 "submission" port so it's not blocked by ISPs), authenticating and then sending the email. The provider will relay the email to its final destination, but your original IP would still appear in headers.
Gmail et al. still add the headers if you send the message through an actual email client though, just not if you send it through the web client. And I'm guessing that the vast majority of email users still use email clients at least some of the time, given that they work much better on your phone and they're the easiest way for consumers to have a backup of their email.
