I see a couple of straw-mans in the reasoning. First is the claim that taxation assumes that working you get paid for nothing - "When I work hard to earn money, I’m not getting something for nothing. I’m getting something for my time and my time has value." - that is not at all the point, the IRS doesn't assume that. You get paid for your time AND your skills AND your willingness to exchange both for value, whatever value you choose to receive.

Which takes us to the barter part of the essay.

There is a major difference between paying someone to do your chores or doing them yourself. What's the difference? The value you are willing to exchange not to do the chores yourself, of course. Be it lack of skills, lack of time or valuing that time spent doing something else more than what you're giving in exchange.

In other words: the IRS recognizes that you own your own time and are free to do with what you want, but when you exchange it for something of value - money, services, goods - you will be taxed on that value.

Not entering the discussion on taxes: both the lawn mowing person and the house cleaning person recognize that there is more value in the other's time for a given task than their own time, if and how much that differential in value should be taxed it's a different conversation.