Auto Post to Facebook/Twitter I've looked everywhere for this functionality and am very surprised that I can't find it...

Is there a plugin that allows auto posting of entry content to Facebook/Twitter/Other social media channel when the user hits Save?

If there isn't, then I'll build one.
I wrote a plugin that posts to Buffer.com (which is free for up to very reasonable levels of activity).  Buffer in turn allows you to post to Facebook, Twitter, LinkedIn, Google+, etc.:

https://github.com/khalwat/buffer

I use it on several sites to do exactly what you describe.