I'm on board with your crusade. It has always driven me nuts that the calendar months don't just track the lunar cycles. No calendar can be perfectly symmetrical but there are better ways to do it than the Gregorian version.

The (sovereign individual) solution is (of course) not being an employee (paid by a month).

The goal is to work on something that scales easily and makes you money even when you sleep.

13 months would mean all monthly tax payments (where I live that's mandatory social+health taxes) would most certainly just add the 13th payment with no other adjustment.