Ok, finally I've found solution! Obviously I found it 0.0001 seconds after I publicly asked this question.
Problem is that "Games" menu in mentioned /home/user/.config/menus/applications-kmenuedit.menu XML file does not contain a <Merge type="files"/> tag, that's why KDE Kickoff Application Menu was ignoring .desktop files if they belong to this category. And every sample .desktop file I've created probably belonged to Games category.
I changed this XML file to be like this:
<Menu>
<Name>Games</Name>
<Include/>
<Exclude>
<Filename>Quake.desktop</Filename>
</Exclude>
<Layout>
<Merge type="menus"/>
<Merge type="files"/>
Now it seems like everything works as expected. I don't understand why it doesn't work by default.