(inverted order of quotes)
The only way you can move an average up is when you have not reached saturation.
I may be wrong but as far as I understand it once you reach Saturation nothing you record over the average shown is counted therefore your average at saturation point remains static or reduces it does not move up again.
Not correct in detail. Some higher scores will count, too.
Imagine scores of 60, 65 and 70 - a 65 average.
Imagine this is a saturated average. Now, any score of 70 or higher won't change anything, 60, 65, 70 are still the valid pool.
But some scores above the average still do count: Any score below 70, e.g. 67 - well above 65 - would modify the pool to 60, 65, 67, resulting in a 64 average. It's one of the "miracles" of the non-linear calculation of the saturated average: Scores higher than the average can cause a drop.
Let's continue to the showdown:
Imagine shooting a 59 and a 62. The pool will become 59, 60, 62, average 60.33. Let's say the threshold is 60, thus any 61 or lower will make it. In this situation, all the famous former scores that built your former averages, 70, 67, 65, 62 in this example, have become irrelevant as well as the intermediate efforts to lower the average to 65, 64, 60.33.
What counts are the 59, 60, 61 only.