A politician wants his kid in a prestigious school. He can use his influence to persuade university administrators to let his kid in. This probably goes on all the time. But this is a separate system of bribery than what we are talking about.
Even in this specific racket a politician grants favors to a middleman who doesn’t directly pay the politician but feeds money to the college admissions scheme on behalf of the politician. The money is laundered through a middleman.
This scandal is probably the tip of the iceberg and if one believes many politicians are corrupt then they likely would have had “opportunities” to take advantage of this system or systems like it.
In the case of a politician it is a two way street. The university trades on the name of the pol just as much as the pol’s kid trades on the university name. The fundraising and recruiting people will be repeating the names “Chelsea, Cuomo, Obama, Bush, Kennedy” etc at every event for years. And it raises (or maintains) the prestige value.
Think about why it makes the news when one of these kids decides to eschew Harvard or Yale and chooses Brown, instead.