Bray misses a crucial political problem, though: the DMCA. Under US law (and similar laws all over the world), telling people about vulnerabilities in DRM is illegal, meaning that a bug in your browser that makes your email vulnerable to spying might be illegal to report, and will thus potentially never be fixed. Now that the World Wide Web Consortium and all the major browser vendors (even including Mozilla) have capitulated on adding DRM to the Web, this is the most significant political problem in the world of trusting your browser.
So I looked around to see if anyone was doing this. Well, sort of. I include the first thing I discovered mostly for amusement value: Signed Scripts in Mozilla, prominently labeled as probably not updated since 1998, and with references to "Netscape Object Signing" and something called SignTool. All it proves is that at some previous point in history, somebody thought this was a good idea.
People still do, in some contexts: Over at Firefox Marketplace there's a writeup on Packaged apps, where all the resources come out of a zipfile. It says: "privileged and certified apps are digitally signed to enable the use of privileged and certified APIs. Privileged apps are signed as part of the Marketplace review process, while certified apps are signed by device manufacturers or operators."
I wondered if there were something similar on the Chrome side. I dug around in the docs and, sure enough, a Chrome extension is signed with the developer's private key. (I couldn't find anything similar for Chrome apps as opposed to extensions, but maybe I was just looking in the wrong place).
Trusting Browser Code [Tim Bray/Ongoing]
(Image: uncle sam wants your privacy, Jeff Shuler, CC-BY)