I've been listening to The Infinite Bad podcast series--which is thoroughly recommended here with special thanks to Candace (aka Ramona Briars) for being its ambassador--which uses a cleverly simple d20 system to run a Cthulhu Mythos style game. I couldn't find explicit detail in how the system is designed, so I built a clone of it that I'm calling (for the moment at least) d20 Pronto. This was my first attempt at a one-page system, but it extended into four pages with example abilities, Game Master advice, and a brief Glossary of terms. It is built as setting agnostic with abundant notes to encourage customization. Incidentally, I am also looking to introduce RPGs to some older players and this system seemed light enough for easy play and deep enough for player customization. And there is nothing like rolling with advantage and seeing two 1s looking back at you or rolling with disadvantage and seeing two 20s.
If you're looking to put your D&D dice sets to work teaching family or little ones, try out d20 Pronto. Characters can be written on notecards or typed up in advance. Its classless system means you can build spellswords, investigatory martial artists, or tech-savvy soldiers depending on your setting. And it all resolves over d20s which are so beloved by many RPG players. There are definitely notes of Powered by the Apocalypse and Call of Cthulhu RPG (How could there not be after listening to Infinite Bad?), but it is intended to lead into other d20 systems with gradually providing elements like Exhaustion, rests, and the like.