I want the ability to add my own picture-in-picture display or overlay of text and other dynamic content.

Example: watching a movie but want the live score of a sports match scraped from a public website to be displayed in a corner.

OR while watching a sports match -- i want a overlay feed of text from a chat stream for a select web source

Looking forward for some public experiments / open projects in this space i could leverage. Dont have the skills to attempt it myself from scratch.

Honestly your best bet is going to be buying a mini PC and hooking it up to any TV of your choice as the only input. Most bespoke hardware is too locked down to make anything like that possible.