This can be done with most templates. There's usually setting that lets you turn OFF the component function, hiding the main body. At that point it's just a page filled with modules. But even if you don't turn that off, you can turn off every module including the menu for the front page menu item and you'll end up with just a blank article. Stick 3 buttons inside that article and you're all set.
I think part of the problem here may be that you need to study and play with Joomla more before assuming things about it. You can't just install a "make my front page 3 buttons" module and be done, you've got plan out the page.
Here's a quick glance at how Joomla works:
Where it says component is the "body" where articles or installed components generate their output.
So either you can shut off the component for the front and just publish a single module with 3 buttons in it, or you turn off all the modules and publish just an article.
The last option is to stick an index.html file on your server in the root level of your public html folder and stick your buttons on that. Most servers will grab that html file FIRST and not show joomla at all until you visit a link you know is in the joomla structure (or link to it from a button).