Solved (I'm really not sure this is the fastest solution but it works)
make a copy of the template folder and rename it as "home-template" (for example)
- rename any occurence of "gk_simplicity" you can find into file templateDetails.xml to "home-template"
- copy in language folder the file "en-GB.tpl_gk_simplicity.ini" and change it as "en-GB.tpl_home-template.ini"
- discover the new template thanks to the Joomla admin tool
- Open it and assign it to home page only
- comment out in HTML/layouts/default.php from line 98 (just after "</header>")
to line 147 (just before "<?php if($this->API->modules('bottom1')) : ?>"
- then copy paste Style from the original template to the copy (fonts, features, advanced, etc.)
Doing that you can have the home page with only the mainmenu, the header module, and the footer