Here, I wasted part of my day so you don't have to. Yes, they have gotten beat by some private schools- in the years that they weren't getting beat by public schools. The MSHSAA site is not wrong- Ezekiel Elliott's senior season at Burroughs was 2012, the only year PHC played him.
2010 Lost to Dexter in District Semis
2011 Lost to Ste. Genevieve in District Semis
2012 Lost to Burroughs in State Qtrs (Maryville beat Burroughs)
2013 Lost to Burroughs in State Qtrs (Burroughs lost to Seneca in semi, who lost to Maryville in title game)
2014 Lost to Burroughs in State Qtrs, (who beat Seneca by 1 in semi, then lost to Oak Grove 49-27 in title game)
**2015 Lost to Burroughs in State Qtrs (who beat Odessa in title game)
2016 Lost to Miller Career Academy in State Qtrs.
2017 Lost to Lutheran South in State Qtrs (who lost to Mt. Vernon, who got lost to Maryville 47-9 in title game)
**2018 Lost to Kennett in District Finals (Trinity Catholic eventual state champion)
2019 Lost to Ste. Genevieve in District Semis
2020 Lost to Cardinal Ritter in District Semis (who lost to Blair Oaks in State semi)
**2021 Lost to St. Mary's in State Qtrs (St. Mary's won state title)
**Private school state champion in 2015, 2018, 2021
Bold type- Park Hills Central lost to eventual state champion
So, like usual, the "who knows what might have happened," (if the evil cheating private schools hadn't beaten us) excuse, doesn't hold up. Other public schools were eventually beating the private schools in all the other years. And PHC lost to the eventual state champ in only 2 years. So, "who knows what might have happened," but...probably not much.