Some context for those interested: Berkeley enrolls approximately 30k undergraduates; the Ivy League combined enrolls approximately 60k. The most commonly cited cutoff for "low income students" is recipients of federal Pell grants, which are (roughly speaking) available to US citizen or permanent resident undergraduates with a family AGI of <$50k.
According to https://www.usnews.com/best-colleges/rankings/national-unive... from US News, 34% of Berkeley undergraduates receive Pell Grants. (Berkeley's own website claims 32%.) This is the second-highest of the top 25 national universities, behind only UCLA at 38%. As a result, there are approximately 10k students receiving Pell grants at Berkeley.
According to the same source, the percent of recipients in the Ivy league ranges from 31% at Columbia to 13% at Yale. Overall across the eight schools, it appears to be approximately 18% and slightly shy of 11k students.
From a preliminary examination, it appears that the claim is indeed true for other UC campuses (of which only UCLA is a top 25 university) but not Berkeley as stated. The UC system makes the claim about five campuses other than Berkeley: https://www.universityofcalifornia.edu/news/half-new-uc-stud...
The claim may have been true five years ago, as reported here: http://news.berkeley.edu/2012/01/17/freshman-apps/
There are two ways to fix this problem--assuming one thinks it's a problem.
One would be to leverage the myriad benefits we provide to do more to ensure that these organizations are acting in the public interest. That's a pretty tall order though. We'd be looking at a ton of new laws/regulations/bureaucracies to make tens of thousands of determinations. In this particular case, it's not clear to me that Harvard isn't at least not acting against the public interest.
The other way would be to eliminate the tax advantages and let the chips fall where they may. Again, assuming one thinks there is a problem, that seems the preferable solution to me.
This is an aside, but I don't entirely agree. The tax exemption Harvard receives on its massive endowment is, per undergraduate, worth much more than the direct government subsidy public universities receive per student. UC Berkeley alone enrolls more low income students than the entire Ivy League combined (this is true of several UCs). Harvard also gets a monumental amount of funding directly from the federal government, and while much of this is in the form of research grants, it is allowed to redirect a substantial portion of this to operating costs.
Harvard gets this government favored status (in many ways, far more favored by the government than a "public" university, which has many more obligations) precisely because it is ostensibly acting in the public interest.