When I log into the Bug Reporter tool that Apple has, I can only see my bugs. I normally like to search the system to see if other people have filed a bug before posting, but I can't see any way to do this. I can only see my bugs, and post new bugs, but I can't see any way to browse or search the whole bug system.
Is this an Apple restriction, or am I missing something?
Nope. You can't. The Apple bug system (Radar) isn't really open to 3rd party developers. It is very frustrating.
As other people have said here, you can't see the bugs that other people have reported to Apple. However, I recently read that Apple uses multiple bug reports for the same issue as an indicator of the severity or priority of a bug or request (see this blog post). So while I generally agree with you that it's better to search first and try to avoid duplicate bug reports, in Apple's case you just file a new bug report every time.
The best we have right now is OpenRadar. Note that it is completely 3rd party, meaning Apple doesn't monitor it and it requires people to submit bug reports twice (once on Apple's Radar and once on OpenRadar).