Seeing as shrew-like animals are the oldest species of mammals and date all the way back to Triassic era, if you killed off one rodent species another would necessarily take its place. They also managed to survive several great extinctions in the last 250 million years and gave rise to our own species. Given the importance of them in the food chain, the world would most certainly be worse off without them.
Man I had a rat infestation in my house recently, and after that I think I could get behind a rat genocide.
Those fuckers got into my pantry and started knocking jars over, they poop on everything, they make a lot of noise, they get into the air conditioning ducts, they will chew through pretty much anything that isn't thick metal or plastic, and they reproduce like crazy.
We were able to get it under control and we hired someone to find and seal up the spots they were getting in from, but I gotta say that it wasn't an experience I want to go through ever again.
The only "good" thing about rats vs mice is that they're decidedly not subtle, so it's a lot more clear that you actually got them all. Mice are smaller and sneakier and when we had an issue with them a few years ago it took quite awhile to know before we knew they were gone.