Whether fighting one particular suid binary worth it, is questionable indeed. But this is good direction. Another modern approach to this problem is run0 from systemd.
As opposed to running background processes as root...?
This is just mindless dogma at this point. You're going to need something to elevate permissions, and setuid is as good of a scheme as any. ssh or run0 are not magic and just as "vulnerable" as setuid or anything else. Any of these schemes are "security holes" if you abuse it.
When there's a daemon running in the background, the attack surface is more commonly understood. The environment is not under attacker control.
Libraries rarely treat data from socket as "trusted" but often blindly trust environment variables, or stdin/stdout/stderr.
sudo etc. already clear much of the environment. And you're going to want to keep some of it because people expect "sudo foo" to work (which you can't do without PATH).
"Fedora 40 Plans To Unify /usr/bin and /usr/sbin" (2024) https://news.ycombinator.com/item?id=38757975 ; a find expression to locate files with the setuid and setgid bits, setcap,
man run0: https://www.freedesktop.org/software/systemd/man/devel/run0....