While you are right in that you shouldn't stay on the same place forever, there is definitely very huge and important lessons in staying at a place long enough that you don't only have to deliver something, but also get to maintain it over time.
I've found I actually got to stay "long enough" to maintain delivered projects much more an independent than as an employee.
As an employee I wasn't able to maintain projects longer than 2-3 years even when I wanted, due to fixed term (academic) employment contracts, redundancy, projects getting terminated due to business decisions, and reassignment to different projects.
As an independent I found I've had very long term ongoing client relationships (7+ years for one, 5 years for another so far), where they call me back to maintain their products for years after shipping.
During that, it's fine and even expected that I have overlapping newer work with other clients.
So I'm expected to maintain a number of things long after shipping, and that's what I expect to be called to do if I did a good job of delivering the project in the first place.
I think my experience of maintenance and dogfooding may be the opposite of what people think of, when they think about employees vs. independents.
