Truth be told, I think everyone, DM and player alike, cares about balance to a limited extent; but the point at which they stop caring is reached and exceeded long before anything like 4e's all balance all the time design rears its head.
I think most players do not think about it at all. Not even 1 little bit. I think about it far more when I am on this forum than I do at any point playing or DMing or even discussing the game.
As long as players fell that what they're playing is playable most of the time, that's good enough.
Agreed but all classes are playable all the time. There are characters that are not playable, but that is because of abnormally poor rolls or really bad build choices (class not being one of those bad choices).
TBH I think there are generally more "bad choices" with strong classes than vice versa. It is pretty difficult to build an unplayable Fighter or Monk as long as you have decent (14+) scores in the important abilities. You can easily build a bad Wizard or Sorcerer by choosing objectively bad spells.
As a DM I do pay some attention to overall long-term balance between classes, in order that they're all vaguely of the same appeal to players.
I don't and even if I did pay attention to class imbalance, the imbalance at the table due to ability rolls, poor build choices or simply bad in game choices would render that concern about class balance irrelevant in play.
Over the past two years, the PCs we have had die in a game I was playing or DMing were: a Cleric, two Barbarians (one was resurrected), a Bard, two Artificers, Two Wizards (one was resurrected), a Monk, a Rogue and a Paladin (resurrected). *
Most of those PCs died between levels 1 and 9. There is not much correlation between class strength and in-game success in that relatively large sample. If class balance really mattered, you would expect the list of characters killed would be dominated by Monks, Barbarians, Fighters and Rogues as they are the weaker classes at those levels and Roges, Fighters and Barbarians are among the most common classes played.
* Note the list does not include characters who went down but did not die and it does not include one Fighter who sacrificed himself at the end of a campaign to close a portal to elemental water from the other side.