Let me take a different tack on this. The statement as presented,
There is no law which requires the average American worker (in the
private sector) to pay a direct unapportioned tax on their labor and
compensation, or services.
is exactly (in a larger sense) correct, but this larger sense does not help the speaker.
The 16th amendment specifically permits an unapportioned income tax.
The Congress shall have power to lay and collect taxes on incomes,
from whatever source derived, without apportionment among the several
States, and without regard to any census or enumeration.
The internal revenue code (US Code, Title 26) is enacted by Congress, imposes the various revenue taxes, and provides penalties for non-compliance. If this is not a law, I don't know what is.
Thing is, an income tax (as described in the OP) is not a direct tax, and hence does not need to be apportioned. The meaning of "direct" is often confused, and the fact that a taxpayer pays "directly" to the IRS is often the underlying source of confusion.
The Constitution provides that
No Capitation, or other direct, Tax shall be laid, unless in
Proportion to the Census or Enumeration herein before directed to be
taken. - Section 1, Article 9, Clause 4.
This refers to a head tax, but there are (in US law) only 3 kinds of direct taxes:
Only three taxes are definitely known to be direct: (1) a capitation [
. . . ], (2) a tax upon real property, and (3) a tax upon personal
property. - Murphy v. Internal Revenue Service and United StatesLink
Direct taxes are taxes "directly" on property. Indirect taxes are taxes on transactions, or transfers of wealth. And most income falls squarely in the latter category.
In fact the apportionment/enumeration requirement is so burdensome that the Federal government has never even bothered to try for a national property tax or real estate tax. In this day of computers and networked data bases it would probably be doable, but not in the past.
Of course, you might ask, why the confusion? Well, that goes to Pollock v. Farmers' Loan & Trust Company, an 1895 Supreme Court decision that held that a tax on income derived directly from property (such as rent or dividends) was in fact a direct tax.
...taxes on real estate being indisputably direct taxes, taxes on the
rents or income of real estate are equally direct taxes.
It made no mention of earned income (as from a job), and so does not apply to what most see as "income tax". The decision invalidated the first general income tax, the Wilson-Gorman Tariff Act of 1894, and promised to really complicate bookkeeping for any subsequent income tax. The 16th amendment was a direct response to this situation.
So, as I started out saying, the statement is precisely correct. There is no law which requires payment of a direct tax (unapportioned or otherwise) on labor, compensation or services. But. The IRS income tax on these categories is an indirect tax, and that tax must be paid. The fact that you will pay directly to the IRS does not make it a direct tax (within this context).