Yeah, and then the player proceeds to roll a '6' and a '3', thus still failing. LOL.
To me... the only way to get the real feeling of "the expert gets to brag" is for the DM to not require a check at all from someone proficient in the check in question. If there's a History question to be answered, all the PCs without proficiency have to make INT (History) checks to remember the answer... whereas if one of the PCs is proficient in History then the DM just gives the answer without even needing to roll. That way there's no chance for the PC to screw it up by even rolling with Advantage.
This is pretty much where I've ended up, too.
3E had something like this with the idea of "trained" vs. "untrained" tasks--some things, if you aren't proficient*, you don't even get to try. No matter how lucky and dextrous you are, you can't pick locks if you don't know how. But there was nothing that went the other way; tasks where the untrained person could roll, and the trained person just succeeded with no questions asked. The latter is much more fun IMO. If you've invested in the skill, you get to step up and confidently say, "I got this." If you haven't, you can still try and hope to get lucky.
The problem with D&D skills is that they lay two whole pillars of the game--exploration and social interaction--on top of a mechanic (the attack roll) that was designed to be one small element of combat. Combat power scales along many different axes: Attack bonus, hit points, number of attacks, damage modifier, special abilities. A combatant's prowess is made up of all these factors multiplied together. So each individual factor must be kept within careful limits. With the attack roll, this takes the form of bounded accuracy.
But when you take the attack roll and make it stand on its own, while keeping those tight limits in place, what you get is a situation where the distinctions between characters are practically invisible. And so the designers put in hacks like expertise, and Reliable Talent, and various ways to get advantage on skill checks, so that the
really skilled characters could stand out at least a little bit. But those hacks weren't available to most characters for a long time, and even now you have to pay an exorbitant price (half of a feat) to give your wizard a modest bonus on Arcana checks.
*In 3E, substitute "having skill ranks" for "proficient."