Why can't I just average the old average with the new number to get a running average?

Checking it against the example, $x = 40$, $\bar{x}_n = 20$, $n = 3$, so the error is $20 \times 2 / 8 = 5$, matching the five the long way round. Two things follow from the formula. The error grows as $n$ grows, so the estimate gets worse the longer the stream runs, which is the opposite of what an average should do. And as $n$ tends to infinity the error settles at $(x - \bar{x}_n)/2$, so the reported figure sits permanently halfway between the old mean and the single most recent value, no matter how many values arrive. It never converges.

The same fault appears from the other direction when two partial answers have to be combined. Suppose one partition has seen 60 events summing to 4200, and another has seen 40 events summing to 3800. Averaging their two means gives 82.5; the answer over all 100 events is 80.

There is a sequential form that is correct, $\bar{x} \leftarrow \bar{x} + (x - \bar{x})/n$ with $n$ incremented first, and it produces the same 25 as the weighted form. It is fine for a single stream where events only ever arrive. It stops being fine the moment anything else is asked of it: it carries no count, so two of them cannot be merged, and a window that moves has events leaving as well as arriving, which needs a retraction that a bare mean cannot supply without the count beside it. Keeping sum and count gives you the mean, the merge and the removal, and costs one more number. Repeatedly applying the shortcut also accumulates rounding error, since each step folds its own error into the state and the state is never checked against a fresh total.

Keep the sum and the count. The mean is then a division at the end, and every update, merge and correction is addition.