What the original poster has described is exactly mean reversion. The mean being defined by the moving average and the thesis being that the prices will revert towards there means (MAs). What you describe is betting on mean reversion of a spread. For example, if you believe that SPY is to high relative to IWM, then you might sell SPY and buy IWM. This is betting that a synthetic asset (SPY - IWM) will revert to some mean (could be an MA). This is commonly referred to as pairs trading, and can be considered a sort of naive statistical arbitrage strategy.
Why can't you use today's closing price? Suppose you are using P = 21. Then for a given stock X(t) at closing time t, the MA is defined as, 1/21*(X(t-20)+X(t-20)+...+X(t-1)+X(t)) Now todays closing print X(t) will only be worth 1/21th of the average. Moreover, if X(t) is approximately equal to X(t-e) where e is small (say 5 minutes, so 3:55 EST) then valuing the averages and determining the basket at 3:55 each day is completely feasible and will be almost exactly the value of waiting for the closing prints at 4:00 EST.
I doubt they don't. If they don't look at your code, why do they care if the algorithm is developed entirely on their platform? But once they know your algorithm, what are the odds you are of any use for them?
If the mean is not stationary, it's not really a mean. The moving average is not stationary, it's a "moving mean". So if you want to refer to it as "moving mean reversion", then I'm with you