You can replace the stile but then you have to deal with painting. Personally, I would just epoxy a patch in. Done properly, the epoxy patch will be stronger than the wood originally was and will hold screws fine.
Personally I use the West systems 2 part epoxy (the 105 resin and the 207 hardene)r and then I typically just add a healthy dose of sawdust to make a putty. However, that is a little expensive. In a pinch I have used JBWeld's woodweld and it works pretty good, again, just make a putty with sawdust and pack it in there and let it dry.
There are many videos on how to do it, but what I would say is this:
clean it good, ensure you get to wood.
apply a good flowing 2 part epoxy as a primer
after the primer epoxy coat is tacky add fine sawdust to 2 part epoxy until you have a putty.
pack the putty in
Let cure overnight
sand smooth and touch up the paint
Alternatively go to grab some Bondo and follow the directions or watch one of the thousands of videos.
Q
Island
Q