Migrate NYT API example to a public API

### MDN URL

https://developer.mozilla.org/en-US/docs/Learn/JavaScript/Client-side_web_APIs/Third_party_APIs

### What specific section or headline is this issue about?

Wiring up the pagination buttons

### What information was incorrect, unhelpful, or incomplete?

The New York Times API example page ( https://mdn.github.io/learning-area/javascript/apis/third-party-apis/nytimes/finished/ ) found at the link “see it running live here” under the heading “Wiring up the pagination buttons” does not show any results or even displays an error message that something is wrong.

**NOTE** - The same applies for the YouTube API example page ( https://mdn.github.io/learning-area/javascript/apis/third-party-apis/youtube/ ).

I'm using Chrome on my Windows 10 machine.

### What did you expect to see?

I expected to see a few articles at least for the search term "food".

### Do you have any supporting links, references, or citations?

no

### Do you have anything more you want to share?

no

### MDN metadata

Page report details

* Folder: `en-us/learn/javascript/client-side_web_apis/third_party_apis`
* MDN URL: https://developer.mozilla.org/en-US/docs/Learn/JavaScript/Client-side_web_APIs/Third_party_APIs
* GitHub URL: https://github.com/mdn/content/blob/main/files/en-us/learn/javascript/client-side_web_apis/third_party_apis/index.md
* Last commit: https://github.com/mdn/content/commit/4def230f85756724b59660e3cd9de363db724ef8
* Document last modified: 2023-02-24T00:05:28.000Z