I'm new to Drupal. As I understand a Drupal(7 in my case) page normally would have layers of wrappers around a block, sections, regions etc ... What I'm trying to achieve is to render some arbitrary content directly into "body" with none of the wrapper overheads, while keeping everything in "head". I've researched hook_block_list_alter which allows you to exclude blocks before rendering - which doesn't offer the ability to customize (removing) sections. Then I came across hook_page_alter looks like a good way to go, although I haven't dug too deep yet, still trying to understand the structure of the $page object. I thought about other options such as using the context module to configure sections/zones when requested page url is in certain pattern.
What's "THE Drupal's way" of doing this?
I would suggest you to first try with Drupal's template system and then, if you can't achieve something try with hooks:
https://www.drupal.org/node/337173
So, as you mentioned drupal has layer templates wrapping one around another. Most outer one is html.tpl.php and if you want something to appear in page head put it there. Then, second inner one is page template, which can be different for any content (node) type. But you also have block templates, field templates...
To override templates for specific content type or field or something else Drupal has some special naming convention. So if you name you template some specific way (and clear the cache!) drupal will start use it in that specific case.
Check the documentation for more details.