That part of the culture WAS consciously cultivated and I found it very valuable. It drew me up and made me more than I was. It was bound up with a culture of open sharing of ideas and cross-training. A notion that you could find compelling work at the company and shift your focus to that work/team.
There was also an effort to encourage new projects and ideas but they didn't go far enough. If I could give one piece of advice it's to create explicit approval and some serious financial incentives for people who start new products at your company. Treat projects like this in the way you'd treat an acquisition of the company making that product. E.G. if your company adopts a side project as a product, give the creators cash, respect, authority, and support to grow it into something great.