I'm confused by Chris and Jamil's embrace of what otherwise seems a practice made questionable in light of current understanding too. Here are the risks of rinsing as I understand them.
Replacing the beer yeast is under with water raises pH which increases risk of infection.
Adding water and/or agitation provides oxygen. This causes the yeast to begin their life cycle, absorbing the oxygen and using up their glycogen reserves. But now in the absence of wort to consume, they are at risk of either autolysis or starvation. And when finally pitched, they will lacked the reserves needed to begin the normal fermentation cycle.
Culture yeast is very precisely adapted, by selection, to this particular life cycle: once exposed to oxygen, they must proceed all the way through growth, fermentation, building reserves, and dormancy, then await the signal of oxygenation to start again. They have lost the genes that allow led them greater adaptability or flexibility to survive an interrupted cycle or to take advantage of food whenever it appears. Wild yeast (and bacteria) are highly flexible, and can take advantage of situations culture yeast cannot. So rinsed yeast, when pitched, is at a distinct disadvantage versus any wild yeast or bacteria present, increasing the risk of infection in the batch on top of the fact that the culture yeast, even if they win, won't necessarily perform at peak.
That last paragraph somewhat explains the mechanism behind the one preceding. A reference on this, which I posted in another thread a while back, is this. https://www.nature.com/articles/d41586-019-00088-y
All in all, yeast wants to live in beer and wait for the next load of fresh wort. Rinsing deprives it of its ideal conditions, increases the risk of poor performance and infection, and is just one more thing to do, when you don't really need another thing to do.
Yeast rinsing seems to have become standard practice in the 19th century when clean water was seen to purify everything, without current understanding of yeast metabolism. While there may be something to the notion (Chris and Jamil note it) that rinsing can separate some petite mutants and bacteria, this should be a relatively minor concern. If it's a concern, reculturing is probably more appropriate.
If rinsing is going to be done, it ought to be done right at pitching, not before storage; in this case, it can serve as a means to aerate the yeast just when needed.
For much of this see Kunze, 18.104.22.168.
Again, you're adding a step, and increasing risk. Why? As to why the AHA site, and all the other books, perpetuate outdated advice, well we've been there before on all sorts of subjects....