How can I browse WWDC session videos directly on my iOS device or AppleTV?
You can find them in iTunes U, and if you log in at 

Apple Developer Site

with a valid developer ID, you will get a list of videos - and there is a button to "View all in iTunes".

When I did this a minute ago on the iPad I got a "One moment please" followed by a redirect to iTunes, and all the videos appeared in categories, all marked "Free".

If it isn't working for you then there must be a problem with your account. If you can access it through the Mac you're probably going to need to talk to Apple about it. 

You've probably thought to try "Clear Cookies and Data" from Preferences->Safari?