I have two catchers that platoon, and the low guy's PC is 62...nowhere close to 37. The better catcher (https://www.whatifsports.com/hbd/Pages/Popups/PlayerRatings.aspx?pid=8840321) is superior in every defensive category and 12 points better in PC. ERA with him behind the plate (93 starts) was 4.24. For the "worse" catcher (69 starts), ERA was 3.65.
Granted, that's only one season. I haven't done any game by game analysis. Maybe my better pitchers pitched a disproportionate number of the lesser guys starts behind the plate. But, this limited bit of data doesn't support the idea that PC makes a big difference.