Another way the rich get in to Ivy League schools is through recruitment for "aristocratic sports" such squash, crew, sailing, polo, golf, etc. Most of these sports are not offered at public schools, and even if they are, the only way to attain the necessary skill level is through expensive private coaching. Golden's "The Price of Admission" provides a thorough look at these practices.
This is the case even aside from alumni-preference. As recently as 2006, 20% of people admitted to Harvard got double time or more on the SAT. In 2003, College Board stopped flagging scores that were achieved under extended time conditions. The way to get double time on the SAT is to get a particular learning disorder diagnosis, something less wealthy students have less access to, particularly considering the lower likelihood of their parents being in the same social circles as local doctors.
Ivy league schools, including Harvard, have always made money off of their ability to mix the paint of the smart with the wealthy and call it the same color. It's always been one way to pass on an inheritance, but in this increasingly meritocratic world, more and more people are realizing that simply attending a school is not much of an indicator of ability or knowledge.
People seem to be answering two completely different questions on that site. The first question is "Does Harvard give preference to legacy admissions?" The answer to that question is unequivocally yes.The second question is "Can you pay your way into Harvard independent of legacy admission." The answer to the second question is far more interesting and far less official.