The things that break quietly
Nothing on a site fails louder than a page that will not load, and almost nothing that matters fails that way. A missing Open Graph image means every share of your launch post is a grey box. An expired certificate takes the whole site down on a date nobody had in a calendar. A robots.txt rule copied from a staging config keeps your best pages out of the index for months.
Each of those is invisible from the page itself and obvious the moment something asks the server directly. That is all these tools do: make the request a crawler, a browser or a social scraper would make, and show you the answer it got.
Check the deployed URL, not the local one
These run against a public URL rather than against your machine, which is deliberate. Most of what goes wrong in this category is introduced between your editor and production - by a CDN, a proxy, a platform default, or a header rule that only applies to one path pattern.
A check that passes locally and fails in production is the normal case here rather than the exception, so the useful test is always the one made from outside.