The term 'meritocracy' entered the popular lexicon with the publishing of https://en.wikipedia.org/wiki/The_Rise_of_the_Meritocracy, a satirical work that lampooned the idea. But the term was adopted seriously and has persisted.
I think it's self-evident that the notion accomplishes nothing, rather just kicks the can to a debate over what you consider meritorious.
I think it's best to focus on two types of merit. There is potential merit ( how much you could be making ) and capturing merit ( how good you are at capturing your potential merit ).
let's take an example. If there are two hotdogs sellers on the street and they sell the exact same hot dogs at the exact same location but one charges two dollars and the other charges four dollars - and the guy who charges more ends up making more money. does the guy who gets paid more have more merit ? yes. they have the same amount of potential merit but the guy who knows the correct price to charge has more capturing merit.
In a competitive free market, people are paid according to their worth. And their worth is what they are paid. Arbitrages are naturally found and eliminated.
This blind, religious-like faith in the "free market" you exhibit doesn't serve to help anyone.
More people sharing a larger total benefit doesn't imply that the average individual receives a larger benefit.
Raising a child alone is a privilege?
I may have taken one more probability class than you, but the expected value for a woman's taxes over her life is 0. There is no such thing as "the average woman", so whatever point you are trying to make is entirely lost.
And women wouldn't have it any other way than to raise a child alone, but that entirely is besides the point.
Curiously enough, that blog post doesn't bother to post the charts of the immensely high market income gap and consumption gap between men and women in New Zealand. Those wholely explain the tax gap, because well, men are earning and spending much more, 50%-75% more for most cohorts.
As for the income support gap, the authors themselves explain that the gov income support is for poorer women who need the money to supplement their own or their spouses' incomes, especially during child-rearing years.
You seem to be correctly noticing some symptoms, but you're misdiagnosing all the problems at hand.