The idea is very very interesting. Drake L. Dragon voices something I have thought about for a long time while playing Civ: who is the player?
I agree that the game is about managing everything, basically, that have to do with the running of a civilization, and does thing quite well, even if there are areas left out (drugs, for one thing) and somethings are in a gray-zone (like the ability for units to survive for 8000 years). Before implementing a feature such as this, I think one have to think about what constitutes the Civ game and where the strive for realism ends.
To be totally realistic, A LOT of things will have to change. First, make the player a line of rulers that begin as the tribe leaders, and which evolve during the game, changing Traits (and introducing a whole new mechanic here with age-dependent Traits) as time goes along, usually by selection, but sometimes there are no choice (only one heir, assassination attempt succeeds and you take control of an line of competitors instead, etc.). Second, that line of rulers should be able to control one city at a time only, until modern times, at least (the capital at first, but would be able to move to other cities at the cost of money and turns) and would have to rely on giving orders for everything else. These orders would have a chance to be carried out depending on things like the leaders popularity and fearfulness, civics, distance to current city and so on. Third, something like this idea would have to be implemented for the civ as a whole, but not only dynamic improvements! Dynamic city founding (based on expeditions, resources, rivers, luck, economy, distance to neighbor, etc), dynamic research (based on culture, civics, resources, the choice of research path (which should be simple: economy, power, religion)) and so on! Also, time is ticking, and the rulers only have so much time, so would have to make choices as to how to spend it.
Thus, the player have only indirect control of a lot of aspects of the game. He/she can still rule the capital, but not much else. Wherein lies the fun in that? Well, the things you CAN control gets much more important. Would you try to influence the research? Or would you rather try to direct your military, so that the soldiers can avoid boredom and beat up the enemy instead of harassing the peasantry? Or would you spend your time hunting, letting the civ take care of itself while you have a good time? Because in real life, unhappy leaders make bad judgements.
Of course, all of this makes the game into a whole other game than civ. Civ is about micromanagement. It is about optimization. I would definitely test a feature such as this, and I would probably like it a lot, but in the end, that would be another game.