If you mean the financially elite, then actually in the US the top private universities are often more affordable for the non-elite. For instance, if your family income is under, I believe, $100k, Stanford waives tuition. If it is under something like $70k, they also waive room and board.
It's actually the top state schools, such as the University of California, that are the hardest for the non-elite to afford (especially for students from out of state).
Also, many of the top private universities don't take transfer students--non-elite students often have to work for a few years before attending college, and it's unpleasant to find out that having tried to take what community colleges you can afford in terms of time and money while holding down a full time job to keep a roof over your head has literally made you ineligible to apply almost anywhere except a state run university (who also tend to heavily discriminate against transfer students).
Even for your example, Stanford, transfer admissions are significantly more selective than normal admissions processes:
"Transfer admission is considerably more competitive than freshman admission. In recent years, the admit rate for transfer students has ranged from 1% to 4%. Between twenty and fifty transfer student spaces are typically available each year."
This further increases the ability of Stanford to keep its student body composed of financial elites--while touting its "generous" financial aid policy. Plus, even an 18 year old applying to Stanford requires SATs, extracurriculars, etc that most of the non financial elite are unable to afford. Most college applications are at $80/pop nowadays; if you can only afford one college application, why pick the university you probably won't be accepted to?
This just isn't true. The student body has a diverse range of incomes, and the financial aid policy is indeed generous sans quotes. The aid packages of Stanford and its peers made them cheaper than in-state tuition minus financial aid at the University of Texas for me.
There is considerably moral hazard in allowing students to omit the financial information of estranged parents. People would "disown" their children to get free tuition.
If you have no family support, you need college more, not less. Of course, places like Stanford are far more concerned with turning a profit than enrolling poor people.
You'll note that the median family income of a Stanford student is $125,000, which is more than double the median family income in the US as a whole. Maybe that's not a lot to you, but it's almost inconceivable to someone who was raised, say, in a family making 12-20K/year.
(You complain that people would disown their children to get free tuition, but forget the fact that before it became profit oriented, Stanford used to have free tuition for everyone--no disowning required.)