a crazy psycho killer is chaotic evil even when hes nice to his family.
a paladin who kills criminals still s lawfull good.
Not crossing the street at a designated crosswalk is a crime. Parking your cart in a way which blocks the street is a crime. A child shoplifting a tiny wooden toy horse is committing a crime.
In the world you and I live in, not all "crimes" are grounds for summary execution on the spot. The same goes for the world of D&D. You need to be very, very careful about what your paladin defines as a crime worthy of death. Without concern for justice, without concern for GOOD, your "paladin" is not lawful good in any way at all, but neutral (or evil). Under your definition, the "paladin" is the same as the crazy psycho killer.
No good DM would let a player at a tabletop game get away with a paladin summarily executing someone for stealing an apple and still claim to be lawful good.
I think this is where video games often fail: the difference between murder and picking up somebody else's property because you knocked it over (yeah, learnt that one the hard way in Oblivion!) isn't exactly nuanced but "criminal is criminal", and more often than not they get treated exactly the same way.
Someone else mentioned that any flavour of lawful is reduced to lawful stupid in such circumstances.
I was going to argue that an Oath of Vengeance paladin can do some pretty nasty stuff to uphold his tenets but I looked it up and there isn't any requirement in 5th edition that a paladin has to be good.
The biggest problem I see is that an action for any one character could shift them towards any alignment depending on their motivations. It's like the idea of going back in time to kill Hitler as a baby. Killing babies is evil but stopping Hitler is good. I psycopath that just likes to kill would be committing an evil act where as someone doing it to stop future suffering could argue that it's good (Although I think somethings should remain evil regardless of intent as it would leave a "stain on your soul"). When you have a thinking DM they can judge your actions a lot better than a static video game.
I still think in a video game setting the best option would be to have a selectable "intent". For example lets say you are given a quest to rescue a noble. Once you get the quest you could have an internal dialog with yourself and be presented with options like:
1. I have to do the right thing and rescue this person.
2. I am obligated the rescue those in distress even if I dislike them.
2. My purse is light, this could be a good pay day.
3. I have nothing better to do, might as well.
4. Nobles taste good.
After you make your selection any alignment shifts related to the quest could be adjusted appropriately. This isn't anywhere near perfect but it could prevent the issue of having to pick dialog choices to keep a particular alignment.
