They are still relevant.
I asked my 2nd grade students to write a 5-paragraph persuasive (opinion) essay, and I realized that most of them didn't know how to properly write a paragraph. No indention. No margins. No focused ideas. Their essays were clustered sentences with scattered ideas. It was like reading my mind.
Although I gave the class a theme--why pizza is the best food ever--there were still some essays that ended up talking about family vacations and other random topics. What? How did that happen? You know how it happened? Because my students weren't clear about paragraphs having 4 or more sentences that are related to a specific topic. In the example of the persuasive essay, there could have been an introductory paragraph;
a supporting paragraph about pizza toppings
a supporting paragraph about the different pizza shapes and sizes
another supporting paragraph about how pizza is inexpensive to buy
and a conclusion paragraph
That would have be a great 5-paragraph persuasive essay on why pizza is the best food ever. Nevertheless, my students couldn't write a great essay without knowing how to write a paragraph. So, I accepted my responsibility in their learning and re-taught the basic fundamentals of paragraph writing...and we went from there.
Let's say you're not an elementary school student and you're not planning on writing an essay, Would you still need to know how to write a paragraph? ABSOLUTELY. You may need to send a professional email, or write a blog post, or send a letter to a creditor, or write a long facebook post. In writing, you're always gonna need to organize your words to make it easier for yourself AND your readers.
A little writing tip: don't make your readers work hard to read your work.
"Keep one idea to one paragraph."
Be brave, writers!