In spite all the states have a probability 1 to be reached infinitely many times, the frequency of each kind of state in the chain is not equal. Some states will have a much higher frequency than the others.
In this case, the states where everyone has a number of coins that is between average-2 and average+2 will be extremely infrequent. They will be reached, but you probably have to wait a lot of time to see them.
But there will be some family of cases that appear very frequently after enough time. I'm not sure if the most common case is
1) Some of the persons has 99% of the money and the rest have a tiny amount of money
2) A few persons have almost all the money.
3) There is some kind of heavy tail distribution were everyone is expected to have some money. If you order them by money you will see something like a wiggly line.
I vote for 1), but I'm not sure at all.
This is similar to the typical thermodynamics problems. Imagine that you have 100 boxes in a line, were you can distribute 100000 "atoms" (or coins). In each step the coins can go to neighbor box at random. This is not the same problem, so it has a different solution.
But it's also ergodic and you can see all kind of weird distribution of the coins/atoms if toy wait enough time.
If you wait enough time you can see for example that all the coins/atom went to the leftmost box and all the other are empty. But this is not common at all.
Most of the time all the boxes will have a number of coins/atoms that is close to the average. (Not exactly the average obviously.)
But this is a different problem with the same state space (100 boxes/persons, 10000 atoms/coins). The rules in the Markov chain are different, so the expected frequency of the states is different.
In the thermodynamic problem, the most frequent states are when the atoms are almost evenly distributed. In the problem of the article, I suspect that the most frequent state is when someone has almost all the money.