Kindle Fire Freetime questions
Unfortunately, i do not believe there is any way, unless you are on a 3rd generation and up version of the kindle fire.

If you do have a kindle fire 3 and above, you could attempt to install google play store unofficially on the tablets, and find apps there that can do just what your asking, including youtube kids for them, and set an age limit on that app.

I personally do not recommend amazon tablets, as they dont come with a google play store pre-installed.

As to this link, it states that you must:


"Go to settings > Security > click on enable unknown sources"
Download Google Account Manager on your device
Download Google Services Framework on your device
"If you have the 2017 Fire HD 8 download" Google Play Store - Fire 8 otherwise Google play store - other
Install apps from .apk in order (Google account manager, Google Services Framework, Google Play Store)


The above method is for a tablet that has not been rooted, if you dont know what rooting is, it probably is not rooted.

I hope this helped