If you argued for hours, the problem is with the player not the rules. Even in the even of terrible, ungodly bad rules the player makes their case and accepts the DM's ruling then moved the eff on. Anything after that is on them and not the rules. At worst they can bring it up again after the game. Keeping arguing is indicative of not liking the ruling, of player entitlement, which wouldn't go away no matter how many disclaimers were worked into the stealth rules.
Stealth explicitly mentions it's up to the DM to make a call. It's right there in the rules. Saying it a second if third time is redundant.
This way DMs can let their players snipe repeatedly from behind a barrel if they want OR give them disadvantage to re-hide OR have them be detected and require extraordinary action to re-hide. They can choose what fits their game. (Or the individual encounter as skeletons might be fooled by a barrel but kobolds might not.)
Stealth is archetypal of there not being "one true way" to handle a game.
It's not a problem with the player, when the player is simply arguing what's in the book by RAW.
There's no such thing that states that stealth rules are in the hands of the DM. That's an assumption you have made based on one specific ruling, and your involvement with 5e through the years. Some of us just started playing the game straight from 3e, which was very much rules not rulings, and had nonexposure at all to the designers philosophy of 5e, and the whole rulings not rules thing. Not everyone reads WoTC blogs or tweets.
Its all washed over now, but it sucked at the time. Its horribly confusing for new players and DMs alike. I don't even care if they don't change the ruled at all, even if they simply fixed the layout that would make things way less confusing.