The comparison might be selective/well endowed institutions vs those that are not, rather than a private/public comparison, as there are very selective publics and not selective privates.
But a simple private-public comparison of low income grad rates is here: http://www.cic.edu/Research-and-Data/Making-the-Case/Pages/Low-Income-Student-Graduation-Rates.aspx
I think what you’re asking though is if high-achieving (high GPAs, test scores, etc) students attending elite schools graduate (on time) more often than those with similar stats who choose a not-selective school like a directional or a lower tier private. Right?
True, which is why an answer to the above question would be interesting to see. We know Pell kids graduate on time more often at elite schools than they do at others, but we don’t know if the same high performing in HS kids do better at one or the other. I haven’t seen a study on that.