Except when you get into the actual, historical Lawrence... he's kind of an ass.
"If I had my way, I would build a lethal chamber as big as the Crystal Palace, with a military band playing softly, and a Cinematograph working brightly; then I'd go out in the back streets and main streets and bring them in, all the sick, the halt, and the maimed; I would lead them gently, and they would smile me a weary thanks; and the band would softly bubble out the 'Hallelujah Chorus'"
Anti-democracy, anti-suffrage, pro-despotism - Lawrence displayed all of the worst sins according to the moral judges of the Twentieth Century... And yet he wrote some of the most insightful, heartfelt fiction ever to come out of England. Not in spite of but because of such a reactionary streak.
Idealists may write inspiring prose, but that same inspiration prevent more nuanced critiques. Idealism handicapped the brilliant economic analysis of Marx and dumbed down the best psychological work of Dostoevsky because it presupposes a happy ending to all this.
Reactionaries like Lawrence don't have that problem. Whatever else you can say about them, their vision is clear - Lawrence wrote the grasping pettiness and existential desperation he saw in those around him every day. The depth of his women characters is not in spite of but because of this low opinion of his fellow man, as it gives him a clearer picture of what it means to be a woman in a society that relegates women to second class citizens.
Two of his novellas display this best - The Fox and The Captain's Doll. The former concerns a formerly independent woman and her lady friend coming under the thrall of an exasperating young man, who casually wreaks havoc on their bucolic life with the assumed privilege of a man over women in turn of the century England. In the latter, the same dynamic is challenged by a Scottish officer's Continental mistress who will not be so easily held down. Though the narrative begins with her, it ends with the titular Captain and his bitterness at not being the center of all things, as Lawrence demonstrates through his contempt for a mountain. A thing much grander than the captain, a direct challenge to his "gigantic littleness" as a critic of Dr. Johnson would put it.
And even in this cruel examinations of humanity, Lawrence never hates his characters. As petty and self-centered as they can be, he still paints them as whole people with just as many virtues and fears. And it's exactly such a sympathy without sentiment that allows for his penetrating insights into the human condition.
Move out from Lawrence, and you see this is actually the norm for the best Twentieth Century and later authors. Take Houellebecq's sympathy for the common German soldier, who would much rather be home in Dusseldorf drinking beer and chasing ladies instead of tromping around occupied Paris, waiting to get his throat slit or slit the throat of some other desperate victim. Or, on the subject of Nazis, here's something from Celine's first novel - "So Alcide had asked to do a double hitch, to stay in Toto for six consecutive years instead of three, for the sake of his little niece... With hardly a thought of what he was doing, he had consented to years of torture, to the crushing of his life in this torrid monotony for the sake of a little girl to whom he was vaguely related."
This is what I mean by the Lawrence Paradox. Authors with the worst politics, with the most animosity towards other people, and they can show more compassion in a rural woman lamenting the death of a fox than can a thousand social realists with their politically correct populism.
And for what it's worth, they also love animals.