What's the point of long/complex sourdough feeding techniques?

Absent a definitive answer here, I'm going to take a stab at this based on some experience I had with difficult sourdough starters. Note that I make a lot of sourdough items but do not do any kind of multi-stage prefeeding ritual.

Sourdough is a culture of multiple types of bacteria and wild yeast. Sometimes wild yeasts do not reproduce rapidly or produce as much carbon dioxide (for leavening) as commercial yeasts. I've definitely had the problem before with a starter that it wouldn't produce anything other than a superdense bread, no matter how long the dough rise.

Most of us keep our starters in the fridge (unless we run commercial bakeries). Thing is, at fridge temperatures, the bacterial culture will grow, albeit slowly, but the yeast will hardly reproduce at all, and will produce no carbon dioxide. So, when you take it out of the fridge, if you use the culture immediately you may have an imbalance of active bacteria but sleepy yeast (this is some inexpert armchair biology here, so a big pinch of kosher salt with it).

You are solving this via the use of 90F water. I solve it by proofing a sponge for 7-12 hours. However, there are clearly sourdough aficionados who feel that this is not sufficient to restore the yeast to full strength and that it needs more time and feeding. The feed, split, and discard routine I believe comes from commercial bakeries that never refrigerate their starters because they use it every day; it is the advice given in the Berkeley Cheese Board's cookbook, for example. Doing this over 36 hours or so could be assumed to get the sourdough into the same state it would have been for a sourdough kept constantly at room temperature and rotated every 24 hours, allowing you to use the exact same rising times/temps as they would use.

Now, back to my all-natural wild yeast starter that wouldn't rise very well. I did not try feed-and-split for 36 hours; maybe that would have made it rise adequately and is the reason for this technique. Instead, I started the starter-making process over and got a better strain of yeast the second time.

FuzzyChef is correct that one reason for a multistage approach is to ensure the starter is operating "at full strength" before mixing the final dough. (And it's also important to note the imbalance of bacteria and yeast mentioned in that answer.) A lot of people don't bake bread reliably every week, and reviving a starter that's been in the fridge for longer can sometimes take a feeding or two. A recipe that suggests doing this may be just ensuring the starter is in good condition.

It's also somewhat unpredictable how starters will evolve over time with lots of refrigeration -- some will build up acid over time and then become sluggish. Some are fine for a while as long as they are used regularly, but then something gets imbalanced. Most recipes that recommend a multistage approach will do a larger dilution of the initial starter, which helps to avoid acidic build-up and make sure the yeast always have plenty of food.

I cannot understand why a recipe would suggest discarding part of the mixture after a feeding like that, though. That's just wasteful, and I'd question any recipe that suggests that. (Most recipes that suggest discarding starter on a regular basis on based on old practices that didn't realize it's perfectly fine to keep rather small amounts of starter reserved in the fridge -- you don't need a lot. I often keep mine in a tiny mustard jar; a few tablespoons is generally plenty.)

If you need to revive a sluggish starter, you may end up with too much to use by the time you've fed it enough to revive it. But as a matter of course, that shouldn't happen with healthy starters unless they've been stored for a long time.

Anyhow, there is one more important reason to do a multistage technique for baking bread sometimes, and that is because it does affect flavor profile and can affect stability of the final dough (along with rise, oven spring, etc.). Some people use "sourdough" yeast to produce all sorts of bread types. If you have a starter that produces a rather sour sourdough, doing a multistage recipe (without discarding any) can dilute the sourness and allow yeast growth rates to overtake the lactic bacteria that produce the sour flavor. If you want to produce sweet rolls or buns, or some other bread type without much "sour" flavor using natural yeast, a multistage method is the best approach. Heavier doughs (with whole grains, or with lots of added butter, sugar, and other ingredients) may also require a strong starter to ensure a full rise.

And while you've had good results without prefermenting the starter, excess acid build-up in a starter over time can deteriorate final dough quality. Aside from flavor, excess acid can break down gluten, inhibit yeast growth (compounding the problem -- you wait longer for the dough to rise, and it just gets more sour with ever less rise), and even lead to a loaf lacking elasticity to rise properly in the oven, perhaps instead spreading sideways or showing little discernible oven spring.

Diluting refrigerated starter and feeding it before use in a final dough helps to mitigate these problems, if you have them. With a healthy starter used regularly, you may not encounter such problems, so there's no difficulty and no need to do multistage builds unless you want a less sour final product or start seeing inferior rising power.