The counter-intuitiveness of the axiom of choice is clouding the real issue here ( I think).  

Suppose there are is an fixed but unknown to you sequence of reals  $x_0,x_1,\cdots$ . Furthermore, the referee assures you that all but finitely many are $0$ (no assumption on how many are non-zero nor which). You may choose a set of terms to eliminate (even all but one) and examine their values. Then you choose some other term. If it turns out to be $0$ you win. It seems that your chances are good (even infinitely good in some sense?) but you gain nothing by what you see. 

Now suppose that there are $100$ people who play this game on the same sequence of reals. They can make a strategy ahead of time but then not communicate. Can we say that there is a strategy which makes them **totally** sure that at least $99$ will get a term equal to $0$? Perhaps. As above, put the terms in $100$ infinite subsequences. Person $k$ will look at all the values in every subsequence except subsequence $k$, then choose from subsequence $k$ a term further along than any non-zero term from any other subsequence. 

But if you  are person $17$ can we say that your chance of getting a non-zero value is lower  than if you were the only person playing? Maybe $\frac{0}{100}=0$. Otherwise it doesn't really matter.

To connect this with the given problem, assume again an arbitrary real sequence and you (perhaps as one member of a group) need to deduce one value based on some or all of the rest. There is no harm in revealing to the referee (after the sequence has been set) the family of coset representatives and the entire strategy. And then the referee can go back to the sequence (or $100$ subsequences) and replaces the values by the difference between their actual and "predicted" values. It shouldn't matter.

In other words, given AoC, we may assume a known set of coset representatives such that every real sequence is is (uniquely) the termwise sum of a coset representative and a sequence with only finitely many non-zero terms. Once we accept that (which is weird but familiar) we may assume wlog that the all zero sequence is a coset representative and that the chosen sequence is from that equivalence class.