I've tried this approach, but I found that most email providers aggressively filter out all incoming messages that don't come from validated sources. No matter what I tried deliverability of emails remained very spotty. I did configure DKIM, DMARC, SPIF, etc as suggested. But minimal improvements. Eventually ended up going with AWS SES.
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I've had surprisingly good luck the few times I've tried it - but using VPSs that were a bit more spendy and not known to be spamhavens helped, I think.
The cost of something like AWS SES or Postmark is so low as to make it almost not worth it, but it worked; not that I'd trust "real" email to it - but I was surprised that Wordpress setup emails were getting through with little config.