Steam already allows you to load in shortcuts for your non-steam games into your Steam Library page--and I think you can actually run the games from there if you like just like you can run any Steam game from there. But only Steam-purchased titles will be backed up on Steam, and of course if you are signing in from another computer then the shortcut won't work unless the local software is installed to run it.
I don't see much point in doing any of that through Steam for non-Steam titles, myself. More trouble than it's worth, imo.