Cite, please.
In 5e, they have also been very clear that they are NOT generally hinging operation of the game on the excruciating specificity of individual word choices. They aimed for more natural language, to enable "rulings, not rules."
I also think a rules interpretation based on wording of an entirely different ruleset is goofy. Lots of people didn't play 4e, or played it little, and won't recall, "Wait, they used wording X there, so wording Y here has a different meaning". Interpretations shoudl be internally based, not external.
So, to prove designer's intent (as much good as that may be) you'd need a statement from the designers about 5e specifically.
I am basing the fact that people who wrote 5e are aware of 3e and 4e D&D when they wrote it.
There are similar abilities that read "on your turn", like barbarian reckless attack. It doesn't work on reaction attacks. Meanwhile, the Rogue is worded to match exactly the changes they made in 4e to Rogues to allow them to use sneak attack on opportunity attacks. When they did that change, as an errata, they explicitly stated basically "this is because we thought it felt better if Rogues where good at sticking you with a shiv if you give them an opportunity".
They did an
intentional change that matches this wording in the years immediately before 5e. They released 5e, aware of how 4e worked. They kept that change, and didn't match the very similar barbarian ability wording that excludes it.
I mean "once on your turn" would make it so that Rogues couldn't sneak attack on reactions. One and done. Or even "once per round" works.
And I'm covering intention. You don't need to understand the intention to read the rule: the plain language of "a turn", as in when a monster goes that is a different turn, is used throughout 5e. Your turn ends, your turn starts, and your turn does not extend over the entire round.
Context does inform intention. I don't imply that all DMs must understand the intention in order to understand what the rule says (that you can do it once per turn), I am arguing that this wasn't an accident of wording by the designers. The fact that a random DM might not be aware of the history is irrelevant to what the history tell us the actual intention of the designers was here.
Feel free to argue that it is
reasonable for a DM to change that rule, thinking that was the intention of the designers for rogues not to be able to sneak attack on a reaction. That is a very different argument than "the designers never intended for a rogue to be able to use sneak attack on a reaction, that wording was accidental". It is very
reasonable for the DM to misunderstand the wording and misread the intention of that rule, I will admit.
It is even reasonable for the DM to decide that off-turn sneak attack makes the Rogue too good at fighting, and simply ban it.
It is even reasonable for the DM to decide that "real reactions" doing off-turn sneak attack is ok, but "fake reactions" like that haste trick don't; that the readied action "transports part of your turn" there, while a normal reaction doesn't.
Of course, in a game with any other characters doing charop, this reaction-sneak-attack trick is one of the few ways for a rogue to "keep up" optimization wise (that, booming blade, and MC crit fishing efforts are the 3 ways I know of).