I would prefer to use a neutral <h> tag, that was proposed in the xhtml2 specs. It always made more sense to let the browser infer the place in the hierarchy.

I've seen this done with one of the existing tags and appropriate nesting. <h1> for masthead, <h2> for major subdivisions in a huge page (essentially sub-mastheads), then <h3> for everything else with styling (and nesting in ToCs and such) being dictated either by how nested they are in <section> tags (or <div>s with an appropriate class). H3 here becomes a neutral header, and h4+ are just not used (nor is H2 in short/medium pages).

How does that help with a document structure if everything on the h3 level is the same?

I wish there was a neutral <h> element that could then be specified at an arbitrary <h~n~> sometimes I have documents that have headings 8 levels deep.

It's worse because you need knowledge of the entire rendered page even when you're creating a subsection that may get templated into the overall page.

It's like if you removed relative paths and working directories from all OSes. A lot of things become really annoying to do.

In practice you're really never going to go below <h3> unless you're writing an actual document, in which case you probably should have a good editor.