Its a concept that just doesn't work for a game where everyone has to be in the spotlight at every moment. If everyone is special all of the time then there isn't really a moment for anyone to shine as an individual.
Combat specialists who "sleep" when the non-combat roleplaying is going on are missing a lot of fun. The mentality of not participating in game events because your character isn't optimized for it is foreign to me.
Its fun for an aggressive outgoing barbarian to try and impress the folks at court. Its also fun for a bookish yet clever wizard to use magic in a scrape that doesn't involve blasting things.
Then how is having productive things to add make things less fun? I've had fun in games doing "goofy things", regardless of whether my character was good at it or not. My little halfling warlock was trying to bluff/diplomacy a captured orc into splitting the loot, killing the party, I just needed to know where the loot was being kept, because I knew a fence ...
When he finally caught on that I was lying, he laughed, so I immediately switched to intimidate, sticking a finger up his nose and asking if he'd ever seen an eldritch blast from that close.
Everyone being good at combat doesn't mean that every combat everyone is going to contribute the same things. Certain monsters have resistances/vulnerabilities, defenses they attack, or defenses they have, etc, etc, etc that makes certain characters shine and other characters get beat up. Depending on the dice, certain characters can be the hero ... for example a bronze warder, which is just plain nasty, became a cake walk due to a combo of a warlock daze (save ends) power and HORRIBLE rolling on my part as a DM. The Warlock got to shine as he dazed an elite monster into being pretty ineffective for the entire fight.
Same group, different day, the wizard single handed changed the way the fight would play out by hitting with the Sleep spell and having an unlucky roll on my part cause the elite to fall asleep. I didn't roll as badly as the other case, but it only took one failed saved for all the martial guys to pile on and coup de gras the HP away. By the time he was awake he was killed before he had a second turn to get up and fight back.
Just because everyone has good combat skills doesn't mean no one shines in combat. Just because everyone has good out of combat skills doesn't mean no one shines out of combat. There are still people with skill sets better suited for certain skill challenges. There are still people with combat skills better suited for certain combat challenges.