Openbabel is not converting Gaussian Z-matrix
Turns out that OpenBabel is sensitive to whether blank lines are really blank lines or lines with space characters only. In my case, this was the blank line between the route and the title line. One can see them by pressing the button with a paragraph sign (¶) on it in Your favourite text editor (Notepad++, or even MS Word or  LibreOffice Writer), or by entering :set list in vim or M-x whitespace-mode in Emacs.

What does OpenBabel require for conversion:


First line of the file must begin with a hash sign (#) – a substitute for the Route line
Next line must be absolutely empty
Next line can contain anything You want
Next line again must be absolutely empty
Then comes the charge and multiplicity
And then comes the Gaussian Z-matrix in either compact or full form.


What doesn't OpenBabel require for conversion:


the file to have the extension GZMAT, it can also be GJF
title line not to begin with the hash sign (#) or any other character
route line to contain actual keywords (if You are not trying to convert them to any other input file format, that is)