I'll chime in with my own little piece of anecdotal evidence here...
Dad and I trapped together (hobby) in the fields and woods in "the neighborhood" of where I grew up in rural Northcentral PA from the time I was 13 until I moved to Wyoming. Dad and my sister continued to trap after I left in 2022 or 2023. Early the following spring, they moved out here as well, and they didn't trap much that last fall. We predominantly trapped land predators and occasionally dabbled in a little water trapping where possible. We never put up record numbers of critters. Our average was a coyote or two a year, 7 or 8 foxes (greys and reds), a dozen coon, between one and two dozen grinners, and the occasional skunk. We were good for a bobcat every year or so, and one year we caught two. We ran a few dozen traps on some local landowners and our own little place most years. Wet, muddy weather would usually lead us to run a little less. We also actively predator hunted. Again, we never put up record numbers. A handful of coyotes/fox a year. We were the only ones that trapped in our immediate area. Our mentor and good friend trapped about 5-10 miles away. Our "turfs" didn't overlap much, but it would be dishonest and unrealistic to say that there weren't places where we were dealing with the same animals.
Since we've left that area, we've kept in touch with a few neighbors and old friends. Several have had issues with foxes/coons over the last few years. Many have even commented since that "they didn't realize just how much we kept the predators in check". As stated above, we never really put up great numbers, but we did dog the hell out of them. Between our other hunting, and our trapping, our scent/presence was scattered around hundreds of acres within the same 3 or 4 square mile area. There's certainly something to be said for pressure and how it affects the behaviors of predators, just like any critter. On years where we took it a little easier on the trapping, we would often get more game camera pictures of predators. On years we trapped more, we often wouldn't see as many predators in the "off" season. They seemed to be a little leerier.
Without getting too much more long winded here, I think that proper predator management does play a role in managing game populations. How significant I think can be very situational. We saw lots of instances where there were lots of predators, and there was also lots of game. We also saw times when there were fewer predators, and not substantially more or less game running around.