SOLVED! Okay, so apparently, under cPanel -> Email -> Authentication, there is a configuration area that says:
Additional IP Address blocks for your domains (IPv4 or IPv6):
We had an IP address there which was the location of our business. However, since I am traveling right now and accessing the internet from a different ISP, my current IP address isn’t on that list. So it looks like it’s blocking certain emails from being delivered.
Once I added my new IP address to that cPanel configuration, now I am getting all the phpList emails.
Whew!