I'm afraid, therein lies the problem.
People with codes of conduct, etc SHOULD be thinking and questioning their own actions.
Lawfuls absolutely have to all the time. In fact, I'd go so far as to say that there tends to be a linear relationship in a lawful between wisdom and self-evaluation. How do I keep to the code? By continually meditating on the code and evaluating my past, present, and future performance on that basis. They've got a map; they've got a guidebook; they are supposed to be constantly referencing it and checking where they are against it. Have I strayed? What do I have to do next?
In theory, a lawful might be naturally lawful so that the code is written into his being in such a way that there is no need to reflect on the code because he naturally knows what to do, but this intuitive understanding is probably more typical of a lawful outsider than a lawful mortal, and in any event all this means is that to play such a character the player themselves must be able to approximate that nature - probably by very intensive meditation on the code.
The problem you get is two fold. Chaotics can generally get away with 'to your own self be true', because the individual is the judge of himself in that belief system. Usually you get lawful stupidity when the player plays a lawful, but plays him just going with his gut as in, "I'M AN AGENT OF RIGHTEOUS JUSTICE! FEAR ME EVIL DOERS!", which typically results in behavior that his neither righteous nor just. Or else, you have a player who is trying to adhere to a code, but has in his head this 5 sentence overly simplistic code that has no relationship whatsoever to real moral codes (which IRL usually require lengthy periods of study and indoctrination to learn) and instead of adhering to some complex set of moral maxims the player adherses to a set of beliefs which are trivially disfunctional. So you get complex codes of honor that may contain summaries of complex ideas like, "Never show cowardice", where in the real code there might be pages and pages explaining what that means, distilled down to their 3 wisdom counter part like, "Never retreat" or "Always seek out hazard." and regardless of the supposed wisdom of the character it ends up being played as a blind 3 wisdom fool.
The crappy Paladin stereotype like Miko is BECAUSE they follow the code like a robot, and that makes for an annoying character, not a hero.
Indeed. And not just a robot; a robot with a super-simplified script - the Roomba of Paladins.
If you want to play a believable lawful character, you are going to need to spend a lot of time mediating over what that character believes because lawful characters are defined by the fact that they define themselves in relationship to external orders. They don't see themselves as in charge of themselves and just 'doing what they want', which is the default mode of play of most players. In fact, most so called lawful characterizations tend to players 'doing what they want' where what they want is actually a self-actualizing force for vengeance, judgment, and violence. Hense, most Paladins in play (IME) are played as chaotic evil vigillantes where the ends justify the means and they are accountable to no one.
Honestly, unless the player is from a social/cultural background that does understand the other taking precedence of the self or has really proven themselves to me, I'd pretty much rather they didn't play a lawful. Most groups trend strongly to chaos in my experience regardless of whats on their sheet, and having lawful written there if anything tends to make it worse. (Lawful as an excuse for REALLY doing what you want without regard to anything.)
These problems have nothing really to do with alignment. I could drop alignment entirely, and the same sort of things would happen (because they come up in systems without alignment in the few cases where you see players in those systems not playing the equivalent of ruffians, vigillantes and bandits). As soon as the player conceives of the character following a code, we are in the same place we would be had we still had alignment, and of course the same natural play style of a group tends to come out.