If this is meant to truly calculate the wage someone needs to live, it should at least attempt to assess transfer payments from the government, the same way it estimates living costs.
I'm not sure what part of "the safety net goes away if you have the living wage" you don't understand. The living wage is the wage that you need. Because you're not getting any "transfers" from safety nets at that point. If we had a perfect safety net you could claim any minimum wage, even $0/hr, is the "living wage". But that's not the point. The point is -- what wage would you need to live completely self sufficiently on your own. In policy, it is used to guide when the safety net is no longer necessary. So, including a decent, morally justified, safety net would defeat the purpose of calculating "living wage". tldr it's "living wage" not "living subsidy".
The point isn't whether the subsidy is for the destitute. It's whether it matters for incomes below the living wage or poverty wage. You can't buy milk and bread with a school voucher and the meager child tax credit is a pittance compared to the cost of raising a kid. It's just not relevant to wages below a living wage. Also, what makes you think the numbers for living wage with children don't include those pittances? Child tax credit would make about a $1/hr difference even if it isn't accounted for. We still have a complete failure of a safety net and a failure of basic minimum wage in the US.