Speaking as someone who does have a Facebook page with over 750 likes, I'm aware of how ineffective business pages are at sharing content. It's important to note, however, that it wasn't always as awful as it is: I remember a time when you could use the site as if your page was your account, complete with its own timeline/feed, which theoretically could've been a compromise to the solution the OP is proposing.
I still stick by a Facebook page at this time though for convencience reasons; besides, there's nothing wrong with sharing posts you originally made on your page to your account.