I think I figured it out. Using a second browser where I cleared the cache, I logged into the Gmail account that I’m using for POP and then used the Allow access to your Google Account page:
https://accounts.google.com/b/0/DisplayUnlockCaptcha
This did the trick and I was able to set Poll via POP3 for email replies in the Discourse settings.
I went through this process earlier while still logged into my personal Google account and the Gmail POP account. The “Allow access…” page does not indicate for which account access is being granted, so even though I had been logged into the POP Gmail account, the permission reset was apparently not being applied to the right account.
So is it better to handle incoming mail on the server rather than letting Gmail handle it? I ask because there are numerous posts by the Discourse team expounding the difficulties with email server setups.