Suppose that Harvard's changes to its financial aid policies don't change the pool of applicants it receives. Then the policy simply amounts to a transfer from Harvard to some upper middle class families who send their children there, and there is no reason for any other college to change its behavior. So the impact of the policy is in the change it induces in the applicant pool to Harvard and the improvements that allows Harvard to make in its incoming class.
Who are these additional applicants? Upper middle class students who previously didn't apply to Harvard because, even if they got in, they lacked the financial resources (or the desire to spend them) to attend but who would attend under the new financial aid policies.
I'll conjecture that very few of these students are now applying to, say, Dartmouth. Compared to Harvard, Dartmouth has about the same cost, a financial aid policy that is no more generous than the Harvard's old policy, and an academic reputation that is no better than Harvard's. If financial costs were a barrier at Harvard, then they were a barrier at Dartmouth as well. So if Dartmouth chose not to match Harvard's new policy, I don't see why it would lose students out of its current applicant pool. And what's true of Dartmouth is true of colleges that are presently even less competitive with Harvard, making the opening to the New York Times article about Dickinson College a bit far-fetched. The article does identify the colleges that will see their applicant pools change:
In the competitive scramble for prestige and rankings, numerous colleges already try to lure some top students away from the Ivy League by showering them with “merit aid” even if they are well off and can afford full tuition.
The price of buying a better class just went up, so these colleges will have to modify their behavior if they still want to compete in this way. Let's assume that the higher prices discourage some (even if not all) of this behavior, generating a better applicant pool at Harvard. The better applicant pool, in turn, means a better set of applicants who do not get into Harvard. Some of the students induced to apply will get the spots that would have otherwise gone to other applicants, presumably of all financial backgrounds given Harvard's need-blind admissions policy.
And where will these newly denied applicants go? To their second-choice schools, which will include Dartmouth in some cases. So one impact of Harvard's change in financial aid on colleges like Dartmouth is that it allows them to also be more selective, even without changing their own financial aid policies.
This impact is offset to the extent that there are students admitted to both Harvard and Dartmouth that would choose Dartmouth if both cost the same but Harvard under its new financial aid policies. If it's just Harvard, then I'll conjecture again that this is a small number of students. If other colleges, very few of whom are more competitive with Harvard than is Dartmouth, follow Harvard's lead, then the set of admitted students who might be lost to their second-choice colleges will go up to the point that it would be difficult to maintain a higher price and the same quality of matriculating student.
So it is not Harvard's behavior that is important--it is the behavior of the large number of colleges that consider themselves competitors to Harvard.