I wish there were a way to extract all of the files in the AVCHD or 'Private' file
That's generally a bad idea. Like Craig says, you should be keeping the Private folder intact and unaltered. The media will usually work better that way.
If you want a way to see the individual clips outside of PP more easily, talk to Apple and get them to fix Finder so it works correctly.
You guys know best. The smartest thing I can do right now is listen to all of you and learn.
If you could get the message through to Apple I'm sure it would be better than having it come from me, with my lack of actual experience along the lines that you see most clearly. From what I understand, I should ask Apple to not keep files hidden in 'encapsulated form' in a 'Private' file but have the contained files available by clicking on the 'Private' file?