You're not "double paying" taxes. The employer portion of your taxes when you're an FTE isn't free money; it's part of your cost basis as an employee, and comes out of your comp.
He wasn't implying it was free money. He simply means you're paying the employer's portion as well when you are self-employed. There is no reason to do this on excess income once you have paid yourself a reasonable salary. To determine a reasonable salary simply examine what you would make if you were employed in that same job function by another company. If you are ever audited you can show you made a good faith effort to determine reasonable salary and you will be fine. Many people set it lower than typical industry rates and are never even audited, because frankly the IRS has better things to do. It typically needs to be dramatically below what you would make in industry for you to be audited. Talk to any CPA and they'll tell you the same thing. Taking excess profits from an LLC as a distribution is an entirely legal and appropriate thing to do.
There is no such thing as "filing as an LLC". When you form an LLC by default the IRS considers the LLC a disregarded entity and will tax you just as if you were a sole proprietorship. (i.e. you will report your income on schedule c) You can however elect to have your LLC taxed as an S Corporation. In which case you have to pay yourself a reasonable salary and then you can take a distribution on any excess income above your reasonable salary. When you take a distribution from an S Corp it is only subject to ordinary income tax. You do NOT pay self-employment tax on distributions. If you don't believe me you can read it on TurboTax's website (https://turbotax.intuit.com/tax-tips/small-business-taxes/ho...) or any number of websites for that matter.