This is so cool! I think this should be a shipped app with every distro intended to be a normal desktop OS.
One feature I would really like to see is direct local importing from your pre-installed linux steam. I think that would be handy.
I also consider this project as awesome 
To your points:
I think lutris is intended to work as much independend of the distro it is running on as possible, so a shipping with every distro is, in fact, point of the makers of the distro
And an automated Import of steam would mean it could grab it from your account. And this shouldn’t be, so the import how it is now is as perfect as it can be 