Page breaks? The web, as rendered in browser from smartphone to TV size, didn’t have a real concept of “pages” for a long time. It’s still not a concept widely used.
But are page or section breaks a “problem”?
I’d argue they are not a problem:
Just create a horizontal rule in Markdown - it’s part of Gruber’s original spec. They do act - and are intuitively readable in plain text as breaks between sections.
What you’re describing here doesn’t sound like essential parts of a document.
Logo or branding items do not contain or add essential information to a document. You can just omit them while retaining a document’s information and meaning. Same is true for headers and footers (unlike, and not to be confused with footnotes!), which aren’t really part of a document’s text and content - but merely provide reference or navigation (e.g. page numbers, chapter titles, etc.), or even brand recognition around it.
They can certainly be useful. But to use the analogy of a book: They are not what the original author writes or wrote - they are what the book’s publisher added around in the publishing process.
They are usually repeated content or can be automatically generated as part of the publishing or rendering process (e.g. dynamically generated numbers in an eBook).
If you want to control the positioning of said elements in a page-based context/rending, Markdown is useless, Rich text (as in .rtf) almost as useless too, HTML and CSS are unreliable and difficult. LaTeX is difficult. The “best” solution, that really provides control over such page-based layout is Desktop Publishing (DTP) application.
Side note:
That may be the beauty (or explain great part of the popularity) of Microsoft Word: It strikes a balance between being a word processor that allows for the creation of structured text-based documents. Coupled with rich text formatting and WYSIWYG desktop publishing features. And all of them are (relatively) easily accessible to users.