Hate to tell you this folks, but when you're engaged in combat you don't get to break it off at your leisure.
If you are winning you can.
Not only that, but combatants don't take turns.
Any RPG I've ever played handles combat with turns. It is assumed that the action is happening non stop, the only reason for turns is so the people who are playing the game can keep the actions straight.
This is one reason why I don't like the psionic feat "Up the Walls" which states that a character can run up and down a wall as long as he begins and ends his "turn" on the ground. It goes on to say that if the character "ends his turn" while on a vertical wall, he falls to the ground. Problem is, as you stated, the characters do not stop moving at the end of their turns. Although as a player your turn ends after a set number of actions, your character continues to move and be active just as if it were real life. That is a poorly written feat because it seems to imply that the character stops moving at the end of his turn (and therefore falls to the ground).
Full turns where one playing piece does everything available to that piece before any other piece can do anything at all.
Again, the assumption on the part of the players is supposed to be that the action is taking place nearly simultaneously and without the pauses between turns that we perceive at the table.
These are things that DnD and many other RPGs have had since their inception. I don't think it is causing them to "fail," if indeed they are.
*IF* RPGs are "failing" it is because of the influence of video games and live action games online. Which, by the way, I've been wondering lately why WoTC isn't working more on that angle.
I think they should create a Neverwinter Nights sort of D20 game... since technological integration is clearly in the future of all RPGs.
if the rules do not allow for an action, or an action by a particular playing piece, then that action cannot be performed
That was much more true of 1e and 2e. Remember the "Don't say no, determine difficulty" discussion in the 2e DMG? 3e is designed to cover just about every possible action you can dream up. If you can think of a reasonable action that isn't covered by the d20 rules, you should write up your own rules for it and sell it on RPGNow, because you've obviously identified a need that isn't being addressed.
Furthermore, that is far more true of video games that RPGs. If the programmer didn't anticipate you doing a particular action, you are not going to be able to do it, no matter how convincingly you argue with the game cartridge.
However, I do agree that there is too much of an attitude in 3e that the rules are the end-all and be-all of game adjudication. I'm the sort of DM who will let players attempt just about any hair-brained action they can think of, just because it is fun (and because I remember the "don't Say No" guidelines in the 2e DMG).