connection to outgoing server smtp.gmail.com fails on iphone5s/iOS7.0.1

I looked over at various Apple Support discussions as well as Google product forums and Gmail discussions. No one seems to have any clue as to this problem. I cannot send Gmails via by the default Mail app in iOS7. Whenever I try to send an email, I receive an error message saying "Cannot Send Gmail. Connection to outgoing server smtp.gmail.com failed", and it offers me options to go into mail settings and configure other smtp servers.

Here are a few relevant details.

1. The exact same settings (username/password/SSL/Port 465) has worked on my iPhone since 3GS and whatever iOS was around at that time.
2. Before anyone else asks the obvious questions, I have tried this while connected to the Wi-fi network both at home and at work, with same results. I can send Gmails using the Gmail app (made by Google) just fine. Every other web-enabled actions work. Heck, even saving the message to Draft works fine - because I can access those draft messages over the web on my iPhone or on my computer. It's just the sending that fails.
3. I have deleted and re-added the Gmail account, to no avail. It is receiving emails just fine, but won't send.
4. I have also shut down and restarted my phone to see if that works, but it didn't.
5. In my phone, the Outgoing Mail Server is set as Gmail SMTP server. However, if I look into the details, I find that while the Server is on, all options including User Name etc. are greyed out, including the Done button.

The Primary SMTP server options cannot be changed, it seems. In the previous screen, there is an option to choose alternative smtp servers, the first option being smtp.gmail.com. I tried setting it up with my Gmail username and password, SSL and a port setting of 587. Mail is supposed to use that server if the primary one doesn't work, but that didn't work either.

Can some kind soul, particularly some kind soul from Apple, help?

Message edited for further clarification