My take is a little different, though I mostly agree.
To start I'll agree with Classic. Even though Checkers has been solved, it hasn't stopped being a competitive game, so lets not even consider games like Chess and Go. now include any double blind games. The problem is, these games generally aren't every friendly and require a lot of knowledge and experience. The reason they're unfriendly is not due to flawless victory strategies, but in that it is very difficult to overtake an opponent who's even mildly better than you.
Randomness makes the difference of skill a lot less meaningful and allow even a weaker player to occasionally be successful. Even among serious players of a game with randomnness, there is the added benefit of randomness introducing new situations that keep a game from being predictable.
While it may seem randomness in some degree is sometimes good, Fischer Chess might help. Fischer Chess is Chess with a randomized back row, with the hopes of removing the huge sequence of openings to remember. While that was successful, the game introduced and unprecedented amount of draws in a game that had already been plagued by them by giving little for experienced players to rely on.
While this wasn't a failure, it really only replaced one problem with another. In truth, the only reason openings are even an issue in chess is because of how obscenely old the game is. Any game of adequate complexity can forgo the need for randomness in it's design -- though again, this adds to the friendliness issue.
Also Knizia's 'quote' sort of bugs me. :(