Word or short phrase for something that constantly monitors

Sentinel

"a soldier or guard whose job is to stand and keep watch."

It doesn't imply infallibility, but it implies vigilance. (And really, your watcher program can't be infallible anyway, can it?)

It is a... monitor.

An instrument or device for continuously measuring some quantity or property (in early use chiefly levels of radioactivity). [OED]

It is usually used for medical devices but not necessarily; and monitors don't only measure quantities but they can monitor certain conditions or properties also.

The term also applies to a person who advises or monitors, and extended uses.

A software or hardware device whose specific purpose is to constantly monitor and restart unresponsive components is called a watchdog. For example, the Linux kernel has support for such devices.

Computers are operated by human-made programs, not on their own. That's why it is not easy to put all those adjectives or phrases before computer.

You could consider using "computer with surveillance program(s)" or "computer with surveillance solution(s)".

"Surveillance" means:

the act of carefully watching someone or something especially in order to prevent or detect a crime

[Merriam-Webster]

Overseer

"The overseer noticed that a node went offline."