Actually, it's not wrong in the sense they are putting it. Browsers have been built on the same architectures since Tim Berners-Lee introduced WorldWideWeb.
IE, we have a piece of software on a local computer that basically uses a URL to fetch a hypertext document and all its associated ressources and then uses the local computer's power to render them on-screen graphically. This is what the video means. Webkit (Safari, Chrome, Konqueror), Gecko (Firefox), Trident (Internet Explorer), Presto (Opera) are all the same as WorldWideWeb and Mosaic were in this sense.
So while you are right that some aspects of WorldWideWeb and Mosaic still exist today, I think that what is wrong with their statement is that they are drawing the line very selectively to try to lay claim to the innovations they are citing. They are also ignoring some major browser enhancements over the years.
AJAX and Dynamic HTML went a long way to change the way browsers handled single pages to avoid the constant re-download aspect and many Web 2.0 apps leverage AJAX to pre-fetch things. The early browsers had no concept of the page initiating more requests on its own after rendering it -- this has been a pretty big leap.
Plugins and extensions to browsers as well as sandboxing required serious architectual changes (albeit, with the exception of sandboxing, those occurred about a decade ago).
HTTP 1.1 added persistent connections to retrieve multiple pieces of content form the same server with a single connection and this required some significant refactoring to overcome prior assumptions -- specifically with regards to concurrency.
The introduction of JavaScript and the newer JS engines along with the dynamic aspects of HTML 5 are non-trivial as well lending to some really incredible web pages that former architectures could not handle (check out VW's site on the 21st-Century Beetle).
So if they are saying that previously browsers just stupidly made requests, received content, and rendered it and Amazon is the first to make a significant impact to changing that, then I disagree. Heck their image scaling stuff is not much different than the gzip encoding that is part of HTTP 1.1 specification -- though a nice improvement since gzip cannot compress a JPEG much further, but server-side rescaling can do that. Many of the things they described have been in Opera Mini for over a year.
I just think they need to give credit where credit is due. Their implementation looks great, but to open with "we're the first to really change how browsers function" is a bit of an exaggeration and fails to give credit where it is due. It sounded to me like some of the selective-point-of-view statements that Steve Jobs would make about Apple innovations. Apple does great stuff, but just like everybody else they have stood on the shoulders of others. I think Amazon is doing some great stuff here, but it is not much different from an architecture standpoint than the iSwifter or SkyFire browsers which render Flash in the cloud and then serve it up to a mobile device which otherwise has limitations.
In many ways, I think what Amazon has done here is take the many great innovations that have been out there and put them together in a package with some of their own innovations to deliver a browser that provides the advantages of all those things with a seamless user experience. It sounds very much like what Apple does when they provide feature enhancements to iOS or Mac OS X that may have already existed on competing devices in some form. Apple may make you wait, but when they put it together they make it seamless and they get it right.
Amazon essentially took lots of one-off solutions to the mobile browsing problem (Opera Mini, Skyfire, iSwifter), added some of their own innovations and put them together in a very seamless package to deliver a great user experience backed by their cloud computing power and limitless network bandwidth. Amazon's should say "it just works" to sum it all up rather than "this changes everything".
