I admit, as someone who has been disabled and unable to work since the age of 22, it is really hard to read this discussion without taking it personally.
I have to assume that in the society you are imagining there are no sick, disabled, elderly, or children. Even then though you have some trouble determining "lazy" players. What if you have a husband and wife team at a camp. And they alternate days when they can gather supplies. Or maybe it's just that one plays while the other is at work, and the other plays on their days off. You now have players who probably hang out in the game together often, but one or the other will not be doing much at any given time. Will they be marked?
Your theory is essentially assuming that the ONLY reason to not gather supplies constantly is just out of laziness. You are also assuming for your players that they are not ok with someone being lazy. You are also ignoring the people behind the characters who have varying time, energy, and interest to invest in the game.
I am curious about the numbers of it all. I mean how many resources of what type would a single player need to gather to take care of themselves for 24 hours, or one game cycle, or whatever you measure for that. How much time in real hours would that take on average for them to do it themselves, and for each individual resource? Is there some sort of scorecard telling you who contributed how much? If so, how are you accounting for the overachievers? If you have 10 people and they each require 50 of a resource, they won't each gather 50. Some will gather 5, but others will gather 200. It also sounds like you expect various people to be in charge of a specific resource instead of everyone gathering a bit of each. So, what if you have food gatherers who can play about 18 hours a day total, but your wood gatherers can only play about 12 hours total? How does the group decide who gathers what? And what do you do about players who don't follow those rules. Say you have plenty of wood, but not enough food and a player comes on and insists on gathering wood instead of food, what can the group do about that?
The scenario you are suggesting has a LOT of variables and social rules that you don't seem to even be considering yet.