All it did was teach kids how to write boring simple sentences to meet the implicit "punctuation quota". An equal amount of content compacted into a couple of complex/compound sentences would actually result in marks off.
1. Start with a hook. Engage the reader.
2. Present the ideas (2-3) that you will be discussing in the essay. Have you ever read a rambling blog post? Yeah, they suck. There's no direction and you don't retain much.
Also, 2-3 ideas in one essay is a great number. There's all those studies that say we can only hold 5 things in working memory at once, blah blah blah. Keeping the essay focused on a few core ideas helps the reader retain them better, and the writer to have a well defined scope.
3. Extrapolate the concise ideas in 2-3 concise paragraphs. Ever read a rambling blog post? Yeah, they suck. Telling students to keep paragraphs in 3-5 sentences helps the essay communicate the ideas in a concise manner.
It also helps block out the text in small visually appealing blocks. Ever read super condensed very long paragraphs? Yeah, they suck. It helps to break up your thoughts with some whitespace. (It's almost like coding benefits from this as well...)
4. Conclude your essay. Reiterate what you wanted to cover. This helps the reader retain the ideas, and it allows the author to tie up the ideas in a nice bow. I love when I finish a book or essay and everything comes together and reaffirms what I've been reading the whole time.
This format is not only great building blocks, but it helps you write larger volumes. If you repeat these small steps several times, you create chapters. If you repeat these steps on a macro level, the chapters tie together into a cohesive piece of literature.
These unnecessary "quotas" may sound meaningless, but a lot of people have thought very hard about how to create basic building blocks writers can follow. These building blocks allow the writers to create concise, well formed arguments. "Boring simple sentences" are extremely conducive to clear and concise writing. I'll take boring sentences that form complex ideas over complex fluff that describes nothing any day. (This is almost analogous to good code design too, weird...)
Maybe it's a matter of opinion, but I believe that the grading process is a sufficient guardrail. When the theories are made rigid, it's usually just to ease the grading. I'm okay with that in the right context. In this instance, however, I think it's self-defeating.
Most people will not become scientists. Those who do can quickly pick up the additional burden of formatting requirements, when necessary.
If you want to teach conformity to arbitrary and seemingly pointless strictures, the rest of the public education system does that already.
I wouldn't say that the purpose of the 5-paragaph essay is to teach clear writing. It serves the same purpose as mandating a format for a conference; no format mandate means you get N formats, which makes evaluating them much harder.
Writing is generally good. Expressing yourself is good. Analysis and critique are good. The artificial essay is useless.