No, would be way too chancy. Impossible to complete. Try this though, my friend and I came up with it. Rooks and queens can skip over 1 piece of either color and take another piece as long as it does not put the opposing king in check, then it's illegal. Otherwise, it has all the same rules. I'll show an example of what I mean.
New type of chess?
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Checkmateibeatu - you raise a moderately interesting question. My guess is the resulting rating would be very low - like in the low hundreds - because there'd be plenty of chances of leaving pieces en prise. What is slightly more interesting, for me at least, is how much the rating would rise as you added simple pruning of 'obviously' bad moves. For example, what would the rating be if all moves that left pieces hanging were removed from the set before the random selection? And what would be the next stage? We seem to be reinventing a rather weak form of computer chess here... 
(AP16 - I have no idea what you're getting at, sorry. I don't see how your new illegal moves relate to randomly selecting from a set of legal moves.)
An on-going game that I have started, be prepared for terrible quality of chess! (The opening moves were determined by the Opening Expolrer):
I know, that's what makes it different. They are illegal moves in regular chess but by doing this, you really have to think about what you move and how you move it. It's not relating back to the original way of the random move generator, just doing what the title suggestings, adding a new type of chess.
I just came up with a new variant of chess where you could take all of the legal moves in the position, assaign them all numbers, and run a random number generator to determine the moves. What rating would you think a player that would play that way be? What do you think of this variant? Comment below!