Youtube web part on SharePoint through SharePoint Framework
Luckily for my goal I can leverage the capabilities provided by Youtube API in order to build my solution, a mandatory requirement is to request an API key, please follow this guide. This is a preview of the final result:
The web part property pane offers the possibility to add your own API key, an optional parameter the “Channel Id” to search specifics video contained in a Youtube Channel and the last permits to select the maximum items to display.

This web part can be enriched with many others interesting features, so your contribution is more than welcome.
My solution is available on github: https://github.com/giuleon/SPFx-react-youtube
Source code of this web part is also available on the official GitHub of Microsoft SharePoint Framework client-side web part samples & tutorials: https://github.com/SharePoint/sp-dev-fx-webparts/tree/master/samples/react-youtube