When I began playing full RPGs with my now 6yo a couple of years ago, I started with No Thank You, Evil!, a game specifically designed to be playable by pre-readers. We moved fairly quickly to Basic Fantasy RPG, but it was so limited in options I felt boxed in. So around 6th level I switched us to the RC.
I absolutely love RC D&D, and it's easily my favorite version of the game. I've embellished it with some modern touches like ascending AC and attack bonuses (my kid's math skills are impressive but charts are clumsy and adding negative numbers might be asking a bit too much), and we started using Advantage and Disadvantage as well just due to elegance and simplicity, but the core is still very much BECMI. Right now we are playing through B4: The Lost City, and probably next session we will take on Zargon - my child is clever and paid attention while watching Clash of the Titans, and is therefore armed with the head of the medusa from Tier 7. I hope we win initiative!
The kid also has big goals, from reaching Name Level and becoming a Barbarian Lord to the decision, earlier today, to become a god/immortal upon reaching 36th level (rolling their eyes with an exasperated "I KNOW!" when I pointed out the need for a sponsor, like that's just another box to check on the laundry list). The weapon mastery rules help make characters feel like they're actually getting better at using their tools, and relatively low hit points mean battles are fast and intense.
And maybe, just maybe, we will go Spelljamming at some point.
Hmm. I rambled a bit there.
TL; DR: the D&D Rules Cyclopedia is the single greatest RPG sourcebook ever printed. It has warts but they are easy ones to work around.