Some hosts block outgoing emails if it looks too spammy. I've seen GoDaddy do this and worse they will often refuse to unblock them. Many web hosting control panels have an email deliverability option that shows what was sent and the status.
It can't hurt to see if your domain is on an email blacklist.
https://mxtoolbox.com/blacklists.aspx
As a general rule, using SMTP is preferred and using an email associated with your domain as the from and reply to emails is a good idea. Making sure you have a valid set of SPF and DKIM records too.
If your host says emails are going out, check spam folders. They may be there.
I would like to see a one click unsubscribe feature in phpBB's emails. It seems to be what is expected these days to help these emails from being flagged or thrown into spam folders.
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