That's exactly how everyone else develops (but using open-source libraries): use a module system and glue together things from previous successful projects (made by other people). Why your own private framework is better than what is out there is a whole different matter.
Picking up bits and pieces from old code isn't a framework. Its a specific tool. Think of Backbone or Ember or Angular as a mechanic shop. Things work a very certain way. The copy and paste code from other sites is a hammer, or wrench. You don't have to use them a certain way, but when you have them, you can use how they are intended.
Unless it solves a specific problem you have, not using any framework will almost always mean more bugs, brittle code, a less consistent API, lots of documentation effort (or lack of) and a much larger learning curve for people other than yourself.
And, btw, it's not private. Most of my code is on github. I just don't try to force it down others' throats.