This has to be very, very context specific.
For a while, someone whom I consider a good friend was the president of an elite university, and my parents had several friends of long standing in that role, one at a super-elite university, one at an equivalent LAC, and one at a large public university where admissions was not so competitive. We also knew a smattering of people in high-influence positions at similar institutions.
Some of them I would not have called about my own child; it would have been a violation of our relationship and what I knew about them. (One, in fact, wouldn’t intervene for his own children, something that still rankles the children a bit 40 years later.) In some cases, there were internal politics to consider, and the president was unlikely to want to be known to be intervening in admissions. In some cases – generally less selective institutions – the president was happy to help, provided the applicant was committed to attending if accepted.
If a major gifting opportunity was involved, that would be a different story. But in most such cases, the president would not be dealing with the admissions office. That is part of what the development office does.