“In other words, if paying taxes is like giving tzedaka. Receiving government funds is like taking from the Tamchui.”
Remember that contributions to the tamchui and kuppah were not voluntary. Rambam ruled that anyone not contributing the assessed amount would be given lashes and have his property confiscated by beit din.
It is not well known, but the English Parliament under Queen Elizabeth I set up a system of support for the poor that was surprisingly similar to the kuppah, with mandatory collection of taxes and distribution to the poor according to need. Every English colony would adopt a similar system; one of the persons who served as a poor funds administrator at one point during his career was George Washington.