Also send alerts to my default contact@chess.com email address.
(In most cases, you will only receive 1 email per login.)
Is this set to default contact or @chess.com? (It should be the latter).
(Is your default contact address the gmail one?)
Preferences -> Account Settings -> Mail Settings:
Have you specified your gmail address correctly? You haven't blocked it?
artfizz, thanks for that suggestion. I did change the e-mail setting, so now it is no longer set to default contact but instead it is set to @chess.com. As for the gmail address, it is specified correctly; if it's blocked, I don't know how to unblock it. I have nothing from chess dot com in my gmail spam folder.
OK, so are any email messages arriving in your your chess.com mailbox yet?
Yes; I just got an alert that it's my turn to move.
Actually, it isn't in my chess.com mailbox but in my gmailbox. My chess.com mailbox still looks empty. And the number of alerts at my chess.com home page has increased, so I deselected the option to have move alerts sent to me.
How ironic!
Assuming it reached your gmail by being forwarded (and deleted) from your chess.com mailbox, it was working correctly - until you decided to stop using it. 
I found, among the Utilities in my chess.com e-mail account, a Start Page that has a summary of my e-mails. By right-clicking on these and selecting "Read later", I've managed to get the alerts at my home page down to one. Another curiosity: this same Start Page indicates I have 91 kb in my Inbox, but no e-mails are visible in the chess.com Inbox.
I'm getting an e-mail alert but no e-mail. I've set my chess dot com e-mail account to forward e-mail to my gmail address. But, when I have an e-mail alert at my chess dot com home page, no corresponding e-mail is visible in either my chess dot com e-mail inbox or in my gmail inbox.