Nah, it'd be more akin to:
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You contribute labor to produce widgets that belong to social stock from the outset (the widget does not belong to you as private property)
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You receive labor voucher for your work which is a claim to a specific portion of consumer goods to satisfy your own needs
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Me, who has a labor voucher from different work walks into a distribution center and takes the widget, and the labor time worth of the widget gets subtracted from my voucher. That portion gets consumed permanently and doesn't go to anyone.
What you described was essentially a scenario where goods were still produced as commodities for exchange (generalized form being capitalism) or where private property still existed. In that case yeah, money remains necessary.
Meso American civilization did have money in the form of cacao beans though, and they were largely producing for exchange (there was even a merchant strata), arguably to a higher degree than even feudal societies in Europe. Money doesn't have to be slips of paper, it can be just another commodity (for the longest time in the western world those being gold, silver, copper, etc).
It also wasn't communistic in any way unless you subscribe to the belief that communism is when government does stuff. The "giving a bit up to the state" a tributary system that isn't unique to Meso America, it's purpose not being to "distribute according to the need" but crisis management and self-stabilization.
Also, while money as a whole isn't natural, it develops naturally as a necessity for commodity production.