This relands the [reverted](https://github.com/flutter/engine/pull/55993) [original PR](https://github.com/flutter/engine/pull/55747) with one important adjustment: if the header is empty and has a label, it is rendered as a heading (`<h1>`, `<h2>`, etc) instead of a `<header>`. This is to be consistent with mobile, where headers are frequently used as headings, and screen readers do indeed read it as "heading". Changing all headers to the `<header>` tag turned to out to be too disruptive to existing usages of `SemanticsProperties.header`.
Long-term, when https://github.com/flutter/flutter/issues/155928 is implemented, we could migrate the framework to use `SemanticsProperties.headingLevel` to communicate that something is a heading, and encourage our users to move from `header` to `headingLevel` as well. After that migration is done, we could make all headers proper `<header>` tags, and not special-case empty headers.
Fixes https://github.com/flutter/flutter/issues/152268
c799bc6196