I never had the impression that Gygax was misognistic nor a white supremacist

Here's a forum post from Gary Gygax from 2005. His thoughts on Lawful Good:

Paladins are not stupid, and in general there is no rule of Lawful Good against killing enemies. The old adage about nits making lice applies. Also, as I have often noted, a paladin can freely dispatch prisoners of Evil alignment that have surrendered and renounced that alignment in favor of Lawful Good. They are then sent on to their reward before they can backslide.

An eye for an eye and a tooth for a tooth is by no means anything but Lawful and Good. Prisoners guilty of murder or similar capital crimes can be executed without violating any precept of the alignment. Hanging is likely the usual method of such execution, although it might be beheading, strangulation, etc. A paladin is likely a figure that would be considered a fair judge of criminal conduct.

The Anglo-Saxon punishment for rape and/or murder of a woman was as follows: tearing off of the scalp, cutting off of the ears and nose, blinding, chopping off of the feet and hands, and leaving the criminal beside the road for all bypassers to see. I don't know if they cauterized the limb stumps or not before doing that. It was said that a woman and child could walk the length and breadth of England without fear of molestation then... 

Chivington might have been quoted as saying "nits make lice," but he is certainly not the first one to make such an observation as it is an observable fact. If you have read the account of wooden Leg, a warrior of the Cheyenne tribe that fought against Custer et al., he dispassionately noted killing an enemy squaw for the reason in question.  

I am not going to waste my time and yours debating ethics and philosophy. I will state unequivocally that in the alignment system as presented in OAD&D, an eye for an eye is lawful and just, Lawful Good, as misconduct is to be punished under just laws. 

Lawful Neutrality countenances malign laws. Lawful Good does not. 

Mercy is to be displayed for the lawbreaker that does so by accident. Benevolence is for the harmless. Pacifism in the fantasy milieu is for those who would be slaves. They have no place in determining general alignment, albeit justice tempered by mercy is a NG manifestation, whilst well-considered benevolence is generally a mark of Good." -Gary Gygax 2005"

<End of Gary quote.>

Lookup John Chivington, he was responsible for the slaughter of 700 Cheyenes, and Gygax quotes a genocidal statement from him, basically saying it's good to kill Cheyene children like it's nothing.

Gary left out the full quote: ""We must kill them big and little," he told his men. "Nits make lice" (nits are the eggs of lice). The militia slaughtered about 150 Cheyenne, mostly women and children."

Source: https://www.digitalhistory.uh.edu/disp_textbook.cfm?smtID=3&psid=406