I had heard of this too, and it was one of the reasons I was extremely cautious when starting to host my own email. To my surprise, I didn't run into a single problem like this.

I lack the expertise to elaborate much, but the gist is that people running their own server were caught by measures that were put in place to combat spam. I recall some providers (AOL?) used to be infamous for silently discarding your mails, which is very aggravating for the involved parties.

You had to jump through certain hoops to ensure that your mail arrived. DKIM/SPF are two such hoops (or related techs, anyway; both have detailed Wikipedia entries). Maybe it's not so much of an issue these days.

This happens as a matter of routine; the biggest problem with running your email server is you are not guaranteed that your messages will be delivered.

Reliably delivering email in todays spam-filled environment isn't guaranteed.

Players like MailGun, SendGrid and Postmark provide such features as part of a EaaS product.