EDIT: You know what, the more I think about this, the more I think that something is wrong. I'll just leave it here anyway, but I'm not sure how much of this is actually right...
Ok let's put this to bed. As I said earlier, the Elo ranking is basically an exponentially weighted moving average. Here's the wiki article on that:
https://en.wikipedia.org/wiki/Moving_average#Exponential_moving_average
The formula is as follows:
You can rearrange this formula into one that is directly comparable to Elo rankings:
St - St-1 = a (Yt - St-1)
The term on the left hand side is the change in Elo rankings. The term in brackets on the right hand side is the difference between Actual result and Expected result*. Alpha is the coefficient used in the Elo rankings (K or K*GD).
So the Elo ranking is a basically an exponentially weighted moving average. If you worked through all the maths, you could probably work out exactly how long it takes for past results to diminish to <10% (say) of the current ratings, that is if you knew what K was in the Elo rankings. Of course the past rankings never disappear -- that's the beauty of exponentially weighted averages, you don't throw any data out.
So yes, the Elo ranking does reduce the weight of each event based on how long ago it occurred. Is it the best system? It's pretty good, sure. But there are surely better ARIMA models you can use, if you want to model it like that. My main concern is that the people looking at these ranking systems seem committed to using some kind of points system (see how they've shoehorned points into Elo rankings, rather than leaving it in the form of a moving average?), which may well preclude them from using better models. If our goal is to make predictions, then we should approach this like we do any set of observations, and just pick the best model for the job - even if it doesn't assign points to winners in some obvious fashion.
Of course if our goal is to make a league table, then you definitely want a system that assigns points predictably...
*-Well, sort of. You probably have to multiply it by some sort of scaling factor, which I assume is baked into "K" somehow. At the very least, the LHS is proportional to the RHS.