Thanks. Some interesting research there.
However, this does nothing to support the idea that generally speaking you're better to leave the starter in for that third time, because even if the quality of pitches gets better, the stats bear out that the hitters' adapting to the pitches outweighs that factor.
To this point in 2025:
3rd time facing hitters by starting pitcher -.757 OPSa
1st time facing hitters by relief pitcher -.696 OPSa