There isn't any other way to read it. There are rounds, and turns within those rounds, if they wanted to limit sneak attack to once per round that is the language they would have used.
That's exactly the kind of thinking I would use to back up what you say. There are specific abilities that say "once per round" or "once on your turn" or "on your turn". If they had meant any of those, they would have said it.
My DM and others ruled differently, so I came here to see what logic and justification others use for their interpretation.
I've clarified it in my house rules it to now read: 'You may perform a sneak attack once per turn, on your turn. Additionally, after your turn ends, you may perform a sneak attack with any one successful opportunity attack (that otherwise qualifies for a sneak attack) you make before the start of your next turn'
I also have a house ruled feat called Combat reflexes (which grants advantage on Opportunity attacks, and a bonus reaction each round that can only be used for opportunity attacks) so the clarification was needed.
Two sneak attacks per turn at 20th is roughly 90 DPR (or around that of an action surging 1H Dueling Fighter), so it more or less balances out.
Since you normally can only make one OA per round anyway, your surplus wording "any one successful opportunity attack" is almost superfluous. (You only get one reaction per round, and it resets at the end of your turn.)