It’s similar for Harvard. Some stats are in the freshman survey at The Harvard Crimson | Class of 2024 By the Numbers. ~72% of non-legacies attended public HSs, and ~39% of legacies attended public HS.
Many parents may believe that attending HW is a safe pipeline to an Ivy, and HW tries to present this image, with publishing the matriculation stats and such. However, whether attending increases a the chance of a particular student attending an Ivy depends on the student. For the majority of students, I expect there is no benefit, and for many students I expect there is a negative effect. However, I won’t go further on a tangent.
What is more relevant to this thread is Chicago shows a different pattern. Attending HW may increase chance of admission to Chicago for typical unhooked students from the HS (may be restricted to just ED/ED2 or similar). I expect a similar statement could be made for several other highly selective private HSs. There was a thread awhile back in this forum at UChicago is Tops at Top Prep Schools in which a poster notes that Chicago has a large number of matriculating students at highly selective private HSs, including HW among others. It may be a pattern among this type of HS, rather than something unique about HW that does not occur elsewhere.