Put it this way. Rich parents tend to transmit advantages to their children in all sorts of ways outside school: home tutoring; good genes; investment in activities and books; social networks; a culture that values learning; and positive attitudes.
The discussion I had this weekend centered around a compliment I was receiving about success - I was adamant that a good portion of any success I have had (or will have) stems from being lucky. I am lucky to have great parents, lucky to go through all the experiences I did, lucky to go to a great school, and so forth. I am lucky my parents taught me manners and cared that I always try my best. The person I was debating said that luck did play a small part, but that it mostly came down to how I seized opportunities. So I'm sure the close to correct answer would be some sort of middle ground.
The rest of the article is interesting, though, since the post addresses the starting advantage that certain children have over others. For example, my family was never fabulously wealthy but my parents sacrificed for my brother and I to go to public school, they bought us books all the time, and were a great support system. This was certainly a nice starting advantage to have when it came to my education, so the post does offer an interesting take on how to "level" the playing field.