Change “Thumbnail”, “Medium” and “Large” image sizes using functions.php?

Those setting are saved in options (see options-media.php for option names) so it is not hard to change them.

However those settings are not meant to be changed by theme and I would strongly recommend against messing with them in such way.

Simply put multiple files is how WP (unfortunately and so far) works with images, and it is up to user how to deal with them and not third party theme.

PS I am assuming you are making theme for public release/selling/whatever. If you are going to use it in some locked down environment where decisions are up to you – then feel free to nuke and use native sizes.