Allowing players to "play what they want" means that all character classes need to be able to handle every role possible. You end up with all the character classes generally feeling the same.
This is true. But only if one of your design goals is isolating players from the consequences of thier choices.
Most games do not have "4 pillars." In Shadowrun your team would be running a mighty risk making a run without a magic-user on board, but if you want to do it, no one will stop you.
In 7th Sea game balance does not come off the rails if no one is playing a Swordsman.
In Earthdawn you do not need an Obsidiman Warrior to survive, however tough one might be.
Yes, basic D&D had 4 base classes. So what? 1ed AD&D had, what, 7? The game was not designed around having all 7 in the party. It wasn't designed around any particular party.
D&D does suffer from the peculiar task isolation of it's class design. In particular healing magic being restricted to Cleric has cast a long shadow over the game. This was not, and please don't kid yourself here, done for balance reasons. It was done for flavor reasons.
D&D is built like a log cabin, whose boughs were stolen from Tolkein, and Howard, and Grimm and
Le Morte De Arthur and the like. Wizards do not heal becuase wizards in Swords and Sorcery fiction were usually the bad guys. Clerics heal because healing miracles were the provinces of Saints. The Thief was his own class because the axe weilding Dwarves hired a specialist in the Hobbit. The Monk was a class because E.G.Gygax like the Kung-Fu tv series.
It's probable that healing as the unique province of cleric is a sacred cow whose time is come. It's been chipped away at for a long time, with the Bard in 3e and the "Leader" role invented for 4e, even though we all know they meant "Healer."
Nonetheless if a party of 5 bards shows up for a grim and gritty war campaign they should get killed. If 5 thieves show up, maybe they'll be able do it sneaky ninja style. If 5 wizards show up they'll be great, until they run out of spells or get run down by cavalry.
The solution to unbalanced classes is not making them all exactly equivilent, it's communication between the GM and the players about campaign expectations, and amoungst the players.
Fighter, Wizard, Cleric, Rogue is the classic D&D party, but hardly the only one.
Ranger, Artificer, Druid and Bard fills all the same roles in 3e, but would be quite different in play.
Heck Totemist, Binder, Factotum and Psywarrior would be 4 do-all classes. Would you call them balanced? Do they slot neatly into 4e's role model? I bet it would be a heck of a lot of fun to play.