Hi Holly, thanks for this well-written piece!
While I agree that you can be a successful freelancer without networking, I think it's a ton easier if you have connections you can build upon.
What helped me the most in becoming a successful freelancer was leveraging my network. My first clients were people I previously worked with or knew from networking events.
Once I built credibility through first projects, I was able to continue with e-mailing and gaining more clients outside of my network.
Also, networking events are a great way to gain new perspectives and ideas. Often, people steered me into direction I'd never thought about before.
In short, yes, success is possible without networking but, a lot easier to reach if you make use of human connection :)