Help! My Gmail address doesn't appear in my stock email app.

[datawalla]

I've had my Note 3 for several months (upgraded from a Note 1) and I've never been able to get my Gmail address to appear in the stock email app. It shows up in the Accounts list and the Gmail app works just fine. Two other Gmail addresses that I have also appear in the stock email app and work correctly. But I cannot get my primary email address to appear there. The link to my Google+ account, which is associated with that email address, appears in the "Me" listing in my Contacts, so the phone obviously recognizes the account. When I try to add it to the email app, I get an "Authentication failed" error every time. This has not been a problem up until now because I'm not a huge fan of Gmail and didn't use the account for much, so I would just open the Gmail app when I needed to (the other two addresses are from organizations that have domain-name email addresses that are really Gmail under the hood). But now, I actually need to use the Gmail account on a regular basis and it's a huge hassle to go to one app for all my other email and then remember to open the Gmail app just to check the one account. When I noticed the problem shortly after I got the phone, I deleted and re-added the address a couple of times, but that made no difference. Since it wasn't an issue then, I just let it drop. Now it is an issue. I've seen plenty of online discussions where people have not been able to get their Gmail to sync properly on the Note 3, but none where they couldn't get the account to appear at all. Does anyone have any suggestions?

By the way, please don't suggest that I just import all my other accounts into the Gmail app. I really don't like the Gmail app, which is one reason I haven't used the account much.