As far as I understood the documentation, I should be able to inline POD everywhere outside here-documents.

What part of the documentation suggests that? It would need fixing, it seems.

To the best of my knowledge POD is only allowed between statements, not everywhere.

Update: perlsyn says

Perl has a mechanism for intermixing documentation with source code. While it’s expecting the beginning of a new statement, if the compiler encounters a line that begins with an equal sign and a word, like this

which pretty much confirms my knowledge