I don't like realism in my games. There would definitely be a rush factor, if you only have a character for so many turns then you have to rush to make them powerful and use them in combat before they keel over.
There are many games that disagree with you... The total war series for one, nobody rushes in these game, and the hero's or Generals age and die. In all depends on how it's handled... if there enough things to level and improve all your hero's, it won't make a difference when they die. It all comes down to the player, ie you, and how the game is handled. If you don't know how to play, you may rush to get something done before your character dies, but if your smart you will use strategy, which this game is, to level all your character, especially ones that will take the throne. There just has to be enough things to help you level your characters. Right now players don't even need an big army, why because there is way to much of an overabundance of hero characters, effectively making big army's useless, depending on the player.
Also it would take a while for characters to die, were not talking 20 or 30 turns... think about it. There are what, 2 - 4 seasons per yer (can't remember if they game goes through all the seasons or not), which means if character were set to die between 70 & 100 years that would be 140 - 280 to 200 - 400 turns before death of old age. In this game that would be a lot of time to level your character.
The entire adventuring tree disagrees with you.
Again it's all in how you play, the adventure tree is there for "Adventure", it increase things for adventuring... but that doesn't mean your need an "entire" army of hero's to go on said adventure.
There is nothing that says something like this couldn't be optional under settings, I just think for the players that want it, it would be a nice thing to have. 