Too many large-scale tests can be a burden. So when asked “Should we write a test for this?”, I would of course respond that “It Depends”. I think it depends on three factors:
How easy is the feature to test?
What is the likelihood of the feature breaking?
What is the impact of the break?
A static link in a webpage is easy to test, but is unlikely to break. If you had contractual arrangements that a ink appears, the impact of a break is severe.
A data centre going down is a pretty serious impact, but not only is it unlikely, but it is hard to write an automated test for it. So here, a manual test is probably a more sensible use of your time.