HTML, like any programming language, evolves and with each new version comes new commands and syntax. However, programming languages need to maintain backwards compatibility as much as possible so as not to break code. It's awful when a new version, or evolution, of a programming language not only brings new commands but no longer allows commands from the previous versions. Unfortunately, it happens... and way too often.
You do not need to use HTML5-specific commands for your HTML file to be HTML5-compatible. However, if you want to declare your file as "HTML5" then it certainly needs to use something that discerns it from a previous version of HTML. Otherwise it's just "HTML".
I can understand the confusion as regards the new definitions if you are coming in from the perception of someone who never had to use the old ones. However, know that we've been demanding these new definitions and possibilities for a very long time as they provide specific definitions and possibilities, especially as regards CSS styling and better defined Javascript functionality.
Why are there <header> tags and <H1>, <H2>, etc. tags?
You're mixing them up.
The <header> tag denotes a HEADER section which appears once, although technically it can appear more than once, at the top (preferably) of a page. The <H1>, <H2>, <H3> ... tags are HEADINGS not Headers. They are generally used as titles above paragraphs, divisions or sections and they have various levels of importance as in a heading, a sub-heading, a sub-sub-heading, etc. Today we use them less often than just declaring classes of something like "maintitle", "subtitle", "title2", "title3", etc....
<header> and <footer> sections are wonderful as they clearly state what they're supposed to be, they can be styled the same on every page of a site, and they can contain several divs or grids or such within them.
<main> is important. There can only be one <main> in an HTML document and it separates that portion from the rest for programmatic control.
It may be "simpler" to remember a minimum of commands. But that's similar to trying to speak a language with a limited vocabulary. Eventually you're going to run in to a situation where you're not going to be able to express yourself properly because you lack the specific words or terminology. Programming languages are similar. You can make something run with a limited, and simple, number and breadth of commands. But when you need to improve upon it or evolve it you need more commands and more specificity.
We all used to have <div class="footnote"> divisions which we managed with our stylesheets and we would use ID="X" to manage individual ones from the general. But we did that because we had no choice, not because it was the optimal way to do things. One of the advantages of a declared header and footer is that any code from outside of the HTML page can control or modify the <header> or <footer> content without having to know if the developer used class="footnote" or class="footer" or class="endnote" or whatever... there are lots of advantages and few disadvantages.
We don't use HTML5 tags and commands to "comply". We use them to take advantage of what they bring to us which we did not have before. Yes we all use our commands slightly differently from others. It's a language. We all express ourselves differently and with a different choice of vocabulary to say the same types of things with our unique individual style.
If you are learning HTML now you have the advantage of being able to learn and use the latest and more specific terminology without having to worry about the older terms which were used before. You can, and should, be coming at the language as a new one. If I was learning English today I would completely ignore learning Elisabethan terminology, even though it would be semantically correct, as I can express myself with all of the wide range of more modern, relevant and applicable vocabulary.
I always look forward to learning a new language, programming or otherwise, to be able to learn new ways of expressing my ideas and designs. Enjoy the learning experience and try out and use as many of the new "words" you learn as much as possible.