some comments
There are a number of reasons why Safari could be failing with a session timeout while other browsers are not that are not bugs in Safari. I would guess most likely there is a problem with your cookie settings or popup blocking.
For instance, if the session is maintained on a remote site via a web bug (single pixel gif) referencing between two servers (an auth and login server) and Safari is blocking cookies of that nature (normally a good thing because web bugs are used by advertisers to track user browsing habits and show GET queries to a remote site) then the login will fail to stick. If the session is created by redirecting to a page which then redirects out (a "waiting for" page which is very common) and that redirect is set with a window.open call instead of a window.location.replace() as is proper, then Safari's popup blocking will break this (again, this is a good thing because this is how advertisers do popover and popunder ads). In the latter case, it would have been nice if it made some popping sound when it blocked a popup (a la Netscape/Mozilla).
As for advantages in OS/Browser integration: look to Windows 98+. For the user, there will be no advantage really, but for the developer, you'd get consistent rendering between applications (like the previously mentioned Help). Also applications such as CSS and HTML editors will be able to incorporate Safari's rendering for free. For instance, the preview pane in GoLive might no longer need an internal rendering tool, but instead just use Safari to do the rendering. Also, you can use it to automate browsing tasks (this really isn't a big deal because MacOS already has cURL and AppleScript, but in Windows, the Win Browser component is very useful). Other places: Apple Mail (or third party mail), RSS Newsreaders, personal organizers and journals, etc. These and more could now handle hyperlinking and rendering in a consistant manner.
Also consider XML and stylesheets (XSLT and CSS). Remembering that XML is the native file format for Mac OS X preferences, KeyNote, the next Final Cut Pro and ostensbly most new Apple Software and updates. Safari has an XML DOM component and something that handles stylesheets. These components can be accessed to reduce the codebase and bugs when creating/managing/rendering files.
Windows also uses the browser component to render the navigation in their desktop. I doubt Apple will go so far since (IMHO) browsing the file system on the mac is much easier and this opens it up for customization (and thus breaks a lot of usability).
Apple doesn't need to put it in the core OS. Already a lot of these features are available right now in the Safari you have installed. This is because of the way the OS catalogs internal frameworks just by the presence of the application in your "Applications" folder (and the reason why some people obsessively "update prebindings"). That's so that you don't need messy installers/deinstallers like Windows. The issue with that right now is that those libraries in Safari beta are undocumented and also subject to change. David Hyatt specifically recommends against using them.
Take care,
terry