That's not true. Reading other processes memory requires root even for processes of the same user.
But it turns out that you are right! Well, at least in modern ubuntu/fedora desktop installs, a few years ago they disabled the ptrace(2) system call. ("The maniacs! They did it!", I'm tempted to cry in anger.) You can still read children processes, but unless you change the yama defaults in the kernel, ptrace(2) no longer works for processes that are not children.
I didn't notice because I surely have changed this default in my old laptop because I use ptrace daily. But you are right that modern linuxes are likely thus crippled.
Still, it seems that an attack model based on defending against a rogue user process is quite doomed. Can't this process change the init files of other programs that will run, even if they aren't its children?
No, they often can't. Things like Flatpak make it so apps have no unnecessary access to the filesystem.