I'm hugely in favor of charging more as a freelancer or consultant, but this isn't accurate. In general, software devs have very low overhead, don't spend half their productive time on marketing and promotion (perhaps they should), and the additional taxes you pay when self-employed in the US are either small (other half of SE tax, max ~8% but usually lower for high-income) or negative (you can write off a lot more).
The reason to charge more is because you're delivering more value, not because you need more money. Clients don't give a shit what you need to make. Charge them based on the value you produce, whether that's 2x or 10x what they'd pay an employee.