That's not as simple as it may seem. I've been running my own mailserver on an alternate domain for decades now and it is one thing to be receiving mail there but to send it is a nightmare. Google, Microsoft etc have made it all but impossible to get through their anti-spam measures to the point that they reject tons of perfectly good mail.

In theory yes, in practice the messages from your random domain way too often end up in spam for Gmail and Outlook users, if they get delivered at all.