Can't answer if you should add them or not...

But if you do - you would get some notifications from Google about that website/domain.

I've only ever seen emails of the "There's an increase in 4xx/5xx errors on site/page(s)"

I also get “there were crawl errors”, which upon investigation are for pages that never existed (and I’ve owned the domain for 20 years, so its not a previous owner/operator thing)