Maybe I'm going crazy, but I can't be the only one who thinks that being prompted to make a decision during the enemy turn is not "disrupting the gameplay" but if anything actively participating in it.
Anyway, this idea comes with a lot of problems. Too prone to human error, to occasional distraction, too reliant on timing (and that's assuming the enemy turn plays smoothly without any noticeable hiccup etc).
It's also fundamentally impractical to use for any reaction that can't be called at will (like, say, Cutting words) but requires specific conditions to use.
Respectfully:
It does make sense according to the RAW.
Reactions and readied actions are part of your turn.
But no, you are not crazy: D&D rules can be very complex, even for veterans of the genre. Plenty of rules, spells, and tactics I have yet to try.
In an online table top session, a DM will manage this by asking the dormant party member to "wake up".
