If we look at opportunities (carries + pass attempt + targets) in the games where he wasn't QB1, here's the numbers per game
2017: 0
2018: 3.19
2019: 3.44
2020: 5.58
2021: 3.69
2022: 7.53
2023: 8.36
Notably Dennis Allen took over for Sean Payton in 2022. Right now Taysom is playing more snaps than ever before and is more involved than he ever was under Sean Payton (with the exception of the few games when multiple injuries had him as the QB - obviously he touched the ball more then)