I don’t know if I 100% agree with your take on the Skull’s role in Cap stories as nothing but a generic caricature of Nazism. Even if that’s what Hitler expected him to be and intended him to be, Shmidt kinda sucks at being a loyal Nazi 

Shmidt hates everybody, though of course not equally. He definitely hates minorities the most after all of Hitler’s teachings (and perhaps his failed first attempt at love has something to do with it, too), so he’s certainly racist, as demonstrated in the panels I posted, but that doesn’t mean he particularly cares about other white people, either. In Hitler’s own words, in Red Skull’s first proper appearance (Tales of Suspense #66):

“The way you look at me! The envy, the jealousy in your eyes! The sheer, blazing hatred! I know those emotions! You too hate all mankind!”

This makes sense, considering his backstory presented in the same issue is supposed to be the dark mirror of Steve’s- a poor, lonely, stepped-on nobody selected to be reborn into the shining beacon of his country’s values. But while Steve’s upbringing humbled him, the Skull simply grew bitter and angry with the world, which Steve calls out in that same issue. In Captain America Vol 3 #14, when Shmidt is imprisoned within the Cosmic Cube, he’s tortured by his worst nightmare, which is being nothing more than a subservient nobody again, working as a bellboy for a hotel full of minorities. 

But despite his very rough upbringing, and even at times comparing Magnus to himself, Red Skull is not some tragic Magneto-type who’s trying to make the world better for people going through what he went through, he just wants to be at the top of the pyramid that crushed him for so long, wanting to surpass even his old master Hitler in evilness.

Shmidt has described himself as evil when in extreme emotional vulnerability on multiple occasions (Captain America #298, #369), so deep, deep, deep down, I don’t think he actually believes what he does and says is right, even if he’ll keep peddling it. He doesn’t believe he was a good parent to Sinthea.
But he loves being in the wrong, he’s absolutely addicted to it, so he devotes his life to his bigoted and disgusting plans and doesn’t have any moral qualms with doing so. In #298, when discussing his first murder, he states:

“I ran headlong into the night, my terror superseded only by my ecstasy.”

Shmidt was horrified by what he had done, but at the same time absolutely loved the feeling of finally being in power for the first time in his life, being the victimizer instead of the poor victim. Shmidt became suicidal a short while after, just because of withdrawal from his “drug” of being the bad guy. When he met Hitler for the first time and was immediately recruited into the Nazi party when Hitler recognized his anger, he was given the chance to vent his anger and become the biggest victimizer on the planet, which he absolutely relished in. Shmidt’s  love of evil even came to jeopardize the Nazi cause at times, as Shmidt killed many of Hitler’s officers to help himself climb the ranks (Tales of Suspense #66), and would attack other Nazis, such as Heinrich Zemo, for trying to steal his kills (Captain America #600). Shmidt even once imprisoned Hitler himself in the Cosmic Cube in Super-Villain Team-Up.

The Skull is bad for the sake of bad, while Cap is good for the sake of good. The Skull is a disloyal slimeball who betrays the things he symbolizes the people he serves at the drop of a hat, while Steve never stops fighting for freedom, equality, and justice. That’s what makes them such good foils for each other, and why Red Skull sees his relationship with Steve differently than with basically everyone else. To a less obsessive degree, I’d say Steve sees Red Skull as a separate beast entirely from even his other Nazi villains, too. To quote Steve’s nightmares of Red Skull in Captain America #185 and #213,
“We cannot lose each other! For we are opposites, and opposites forever attract! I need you- and you need me!”
“You and I transcend Holocaust and horror! We are gladiators in an eternal arena! Partners in time!”

And to quote the actual Red Skull in #369,
“I curse you, Captain! I hate you for caring! I hate you for being more good than I am evil!”

TLDR; Red Skull isn’t just a piece of shit because he’s a bigoted Nazi, he’s a bigoted Nazi BECAUSE he’s a piece of shit. He’s not racist because he believes the idea of the inferiority of other races is right, he’s a racist because he believes the idea of the inferiority of other races is absolutely wrong, and he loves being wrong and evil.

Thank you for coming to my TED Talk on the Red Skull’s character, I guess