PagerDuty engineer here. There is much to discuss with respect to "Websites of all sizes are designed to largely self operate with no manual intervention for decent periods of time." The phrase "decent periods of time" is doing a lot of heavy lifting here.
You also said "Otherwise all large websites would be down during holidays, etc." What makes you think large websites don't go down in some fashion or another during Holidays? Are you thinking that there would be nobody to handle incidents over a holiday?
Because every large website has people on call over holidays, and they handle incidents that would become extended outages if left unattended. Many large companies publish incident reports, and if you care to review them, you may find that indeed, their sites do experience partial outages all the time, with no exceptions for holidays (full outages are rare, of course).
Many companies do have "skeleton staffing" during major holidays, so they try to mitigate the risk of a major outage by instituting "code freezes" and not shipping new features.
That doesn't speak to how resilient those sites are, it speaks to the fact that yes, it's a normal things for sites to go down, and yes, humans are needed to keep them up. This whole thing speaks to the difference between "robust" and "resilient."
Robust: The site doesn't go down.
Resilient: When the site goes down, it comes back up with no or little impact on the user/customer experience.
Robust is an architectural property. Resilient is both an architectural and organizational property. Cutting staff impacts resilience, but not robustness. But for most users, the difference between them is invisible. If you see a site that appears to remain up over a holiday, it may be resilient rather than robust.