Explanation for newbies: setuid is a special permission bit that makes an executable run with the permissions of its owner rather than the user executing it. This is often used to let a user run a specific program as root without having sudo access.
If this sounds like a security nightmare, that’s because it is.
In linux, setuid is slowly being phased out by Capabilities. An example of this is the ping command which used to need setuid in order to create raw sockets, but now just needs the cap_net_raw capability. More info: https://unix.stackexchange.com/questions/382771/why-does-ping-need-setuid-permission. Nevertheless, many linux distros still ship with setuid executables, for example passwd from the shadow-utils package.
From what I’ve read, no. Though it doesn’t solve the fundamental problem of a root process handling untrusted input from a regular user.
The TTY method is IMO better as it ties privileges to a piece of physical hardware, bypassing the complexities of userspace elevation of privileges.