This is all a stupid smokescreen for the fact that the character you're describing, is still, in D&D terms, an Evil character. Because once again, alignment measures personal morality on an objective scale. This isn't the Good Place. You can spit out nonsense like "erroneous" and pretend I said stuff I didn't like "all deviations are disallowed", but you're never going to get around the fact that the character you're describing is Evil.
Now he may feel he does a lot of "good", but it's small-g good, he is capital-E Evil, because he is totally amoral and ruthless, and has absolute faith in his personal ability to change the future and always make it "better".
This is a classic villain.
So what are the alignment criteria? Lets use 5e's. From the 5E SRD
Lawful good (LG) creatures can be counted on to do the right thing as expected by society. Gold dragons, paladins, and most dwarves are lawful good.
Neutral good (NG) folk do the best they can to help others according to their needs. Many celestials, some cloud giants, and most gnomes are neutral good.
Chaotic good (CG) creatures act as their conscience directs, with little regard for what others expect. Copper dragons, many elves, and unicorns are chaotic good.
Lawful neutral (LN) individuals act in accordance with law, tradition, or personal codes. Many monks and some wizards are lawful neutral.
Neutral (N) is the alignment of those who prefer to steer clear of moral questions and don’t take sides, doing what seems best at the time. Lizardfolk, most druids, and many humans are neutral.
Chaotic neutral (CN) creatures follow their whims, holding their personal freedom above all else. Many barbarians and rogues, and some bards, are chaotic neutral.
Lawful evil (LE) creatures methodically take what they want, within the limits of a code of tradition, loyalty, or order. Devils, blue dragons, and hobgoblins are lawful evil.
Neutral evil (NE) is the alignment of those who do whatever they can get away with, without compassion or qualms. Many drow, some cloud giants, and goblins are neutral evil.
Chaotic evil (CE) creatures act with arbitrary violence, spurred by their greed, hatred, or bloodlust. Demons, red dragons, and orcs are chaotic evil.
RAW the PC described would be, most likely, Chaotic Good. A PC who acts as his conscience directs, with little regard for what people who dont know what they are talking about and just making stuff up on the internet expect.
In this case, the PCs conscience directs them to kill the prince in order to stop 10,000 deaths. They don't care that you haven't actually read the rules but are pretending to have done so. Their action they are taking is straight up murder so perhaps you may argue they are not lawful, and I will give you that. But there is NO support for your argument here in the actual text. In their hearts and their minds, this PC has decided to take an action that they believe is right. This is chaotic good. Through RAW.
Stop making stuff up.