dnf provides \*bin/docker-init doesn’t return any packages that provide docker-init and the only package in the repo that seems to include support for the docker init is oddly podman-docker. It looks like the docker shipped in the Fedora repos doesn’t support docker init AFAICT. You might need to install it from the upstream docker repos or try podman instead.
The --init flag should work in Fedora by default with moby-engine package. Double check that you have tini-static package installed (should be installed as an dependency of moby-engine) and that the dockerd daemon is started with param --init-path /usr/bin/tini-static (systemctl cat docker).
You can test with command docker run -it --rm --init fedora bash