New template lookup order also with _partials possible?

NO.

Example 1

Given the example lookup order in the docs:

  1. layouts/_partials/footer.section.de.html
  2. layouts/_partials/footer.section.html
  3. layouts/_partials/footer.de.html
  4. layouts/_partials/footer.html

And this call:

{{ partial "footer.section.de.html" . }}

If the de language has not been defined, Hugo ignores the language designator, and layouts/_partials/footer.section.html will be selected.

Example 2

Given this _partials directory:

layouts/_partials/
├── foo.bar.html
├── foo.html
└── foo.oops.html

And these calls:

{{ partial "foo.bar.html" }}
{{ partial "foo.html" }}
{{ partial "foo.oops.html" }}

If the bar and oops output formats have not been defined, Hugo ignores the output format designator, and layouts/_partials/foo.html will be used for all three.

I’m sorry you find this documentation insufficient:

Unlike other template types, partial template selection is based on the file name passed in the partial call. Hugo does not consider the current page kind, content type, logical path, language, or output format when searching for a matching partial template. However, Hugo does apply the same name matching logic it uses for other templates. This means it tries to find the most specific match first, then progressively looks for more general versions if the specific one isn’t found.

Perhaps someone with a better command of the English language can make a suggestion.