Wanting to do an update to this. I'm beginning to think that it is "not" Outlook or MS in this particular case.
On my mobile phone I use a client called Edison Email. Over the past couple days I got 'notifications' of new email in my gmail account. But when I open the application up to view them within the app, there is no email listed. Today is 9:50pm Thursday. The last email I have listed is from 12:52pm Tuesday. Even doing a forced refresh won't update the mail listings.
However, if I go to mail.google.com there's all the emails from the past couple days shown clearly.
So apparently there is an issue with Google Email Services.