Because the plugin in the codepen is a “premium” plugin that uses a cloud service to perform the checks, similar to point 4 in the list above.
I would strongly advise against that. Not only do you have to query your external API (which is rate limited on glitch.me), but each check would have to perform at least a HEAD request against each single resource (i.e. link).
Edit: An intermediate approach would be to create a checksum (hash) of the links that you pass to your link checker API, and then aggressively cache the result on the server side using the checksum as cache ID.