I may just have figured out a fairly complicated way to fix this issue, or at least glean some insight into the problem (as in the course of this writing the problem came back for me). I'll try to explain it in simple terms here, but please ask questions if you don't follow me (and know that I'm writing this after just minutes of not seeing the dreaded password prompt.
Thinking of this possible fix started with first seeing the syncing work with the trial versions of BusyCal and BusySync… So it is not directly an issue with CalDAV on the Google end, and might well be Calendar specific (maybe not OS X Calendar specific if iOS Calendar shares the bug).
Then I paid close attention to the first part of the log entries most of us have posted: “8/12/12 4:58:21.263 PM CalendarAgent[431]:”, specifically “CalendarAgent”. The process that actually does the syncing with Google Calendar is not Calendar directly, but Apple’s system-level background process CalendarAgent. CalendarAgent is located (as found using Activity Monitor in /Applications/Utilities) in /System/Library/PrivateFrameworks/CalendarAgent.framework/Executables.
Then, where does the system (supposedly, given these issue…) store the account passwords? In the keychain, which we can view and edit using Keychain Access (in /Applications/Utilities).
I just did two things that (for more than one functional sync at least) eradicated the password prompts. It may be worth trying if you’ve had these issues, it may work more permanently for you, or at least give you ideas for future attempts at fixing the problem and further insight into the issue. I made two changes to every Google-related keychain item in my login keychain:
1. If the stored password was my old password I updated it to my current/new password.
2. Added CalendarAgent as a process that can access the password under the Access Control tab of the keychain item editor
That’s all I did, and at least for a bit it appeared that I had fixed the issue. I suggest you try this and see if it fixes the problem for you. If not then at least this write-up furthers our documenting of this issue even if it doesn’t resolve it.
Also, anyone and not just registered developers can log in at https://bugreport.apple.com and submit bug reports. I think as many of you who are willing ought to at this point regarding this problem.