Basically, the SUID bit makes a program get the permissions of the owner when executed. If you set /bin/bash as SUID, suddenly every bash shell would be a root shell, kind of. Processes on Linux have a real user ID, an effective user ID, and also a saved user ID that can be used to temporarily drop privileges and gain them back again later.
So tools like sudo and doas use this mechanism to temporarily become root, then run checks to make sure you’re allowed to use sudo, then run your command. But that process is still in your user’s session and process group, and you’re still its real user ID. If anything goes wrong between sudo being root and checking permissions, that can lead to a root shell when you weren’t supposed to, and you have a root exploit. Sudo is entirely responsible for cleaning the environment before launching the child process so that it’s safe.
Run0/systemd-run acts more like an API client. The client, running as your user, asks systemd to create a process and give you its inputs and outputs, which then creates it on your behalf on a clean process tree completely separate from your user session’s process tree and group. The client never ever gets permissions, never has to check for the permissions, it’s systemd that does over D-Bus through PolKit which are both isolated and unprivileged services. So there’s no dangerous code running anywhere to exploit to gain privileges. And it makes run0 very non-special and boring in the process, it really does practically nothing. Want to make your own in Python? You can, safely and quite easily. Any app can easily integrate sudo functionnality fairly safely, and it’ll even trigger the DE’s elevated permission prompt, which is a separate process so you can grant sudo access to an app without it being able to know about your password.
Run0 takes care of interpreting what you want to do, D-Bus passes the message around, PolKit adds its stamp of approval to it, systemd takes care of spawning of the process and only the spawning of the process. Every bit does its job in isolation from the others so it’s hard to exploit.