A nice idea, but probably not a good one.

The problems Tor would bring users in terms of slow connection, risk from malicious exit nodes etc. would outweigh any benefit for most people. Combine this with the fact that to gain anonmyity from Tor you'd need to (at least) disable JS and session cookies, which many ordinary users will consider to break their browsing experience, and you'll find that really all this idea would do is put a lot more load on the Tor network. That's assuming a 'default on' option.

Perhaps an 'anonymous mode' version of 'private browsing' which switches the browser to a more secure configuration (like the Tor browser) and proxies through Tor would mitigate some of these problems.