I don't think pain should be compared. Both these characters went through a lot, and to downplay either side isn't fair. But if we were to compare, this is my overall view of it.
Itachi's pain involved losing more people, as he had to murder the Uchiha clan in order to stop a war from breaking out, including his parents. He's an introverted character, so he was definitely the kind of person who bottled up his pain and didn't have anyone to console with, and at the end of the day, he couldn't tell his own brother that he loved him, and had to pursue that whole crazed maniacal brother persona. Despite Itachi's tragic past, he still grew up with a frame of reference for peace. He had a father that was tremendously proud of him, an affectionate younger brother and girlfriend, and he grew up in the relatively safe Hidden leaf village.
Nagato grew up in a full-blown war and in complete poverty, he had no frame of reference for peace. His parents were murdered infront of him when he was only 7 years old, compared to Itachi's age of 13 when he murdered his clan. Nagato's pain was a lot more physical than Itachi's, in the sense that he had to endure the elements, weaponry and explosions from the war etc. but Nagato still endured mental turmoil as well. Nagato at first managed his pain in a healthy way by creating a group with Yahiko dedicated to peace, and followed Jiraiya's teachings. But if you saw the group you spent over a decade of your life developing for the sake of peace reduced to nothing by Hidden Leaf ninja, the same ninja who murdered your parents, you'd experience deep pain as well. Nagato's hatred for the Leaf is not unwarranted.
Before Kakashi graduated to Genin at age 5, his dad committed suicide. Why? His dad chose to save his comrades instead of completing a mission.
Does Kakashi decide he wants revenge on a village who turned their backs on him, even those he saved?
No, Kakashi decides that the best he can do is learn from his father's mistakes and live his life by the Shinobi rules.
Fast forward to age 12-13, Kakashi is a Jonin and created the new but incomplete Chidori. He now leads his team of Obito and Rin on their first mission under his leadership. When Rin is captured, Kakashi decides to focus on the mission, leading Obito to abandon him to save Rin.
Later, because Obito isn't at a level where he can remain undetected to an enemy, his element of surprise is gone and Kakashi is forced to save him. In the bout, Kakashi loses his eye. If Kakashi had gone with Obito from the start, they may have retained their element of surprise and he may have kept his eye.
But since he chose to value the mission first in his initial decision, Kakashi's eye proves a liability. After a fight, their opponent collapses the cave around them. A rock strikes Kakashi on his blindspot (which he onlygot because he decided mission came first at first). Obito is forced to save him and get crushed.
So because Kakashi didn't decide to help Obito from the start, Kakashi ended up being the liability that caused Obito to lose half of his body.
Then Obito gives Kakashi his Sharingan, and makes him promise to protect Rin. Kakashi promises, and with the Sharingan completes Chidori. So the Chidori represents himself and Obito.
However, Rin is captured and is forced to become the 3 Tails Jinchuriki. Kakashi does his best to keep her safe, but Rin kills herself. On what? Chidori. The jutsu that Obito helped Kakashi complete. The jutsu that represented his friendship with Obito now is the same jutsu that broke his promise to Obito.
Kakashi trucks along, depressed but doing his duty.
Later, the 9 Tails attacks Konoha. Kakashi knows he could help. Kakashi is a Jonin and one of Leaf's most talented even though he's young. But the higher-ups don't let him participate. In the battle he was so close to yet unable to participate in, his Sensei died. And the woman he was tasked to protect for 9 months died as well.
Kakashi went through all of this. And he never became the villain of the series. Even later when Obito is discovered to be alive, Kakashi finds it in himself to kill his former friend because he knows that's what needs to be done. Kakashi struck Obito to kill (twice).
He's just an absolute beast of a character in terms of emotional endurance.
Kakashi has been through some shit for sure, but also seelentau's timeline points out that Kakashi was older than 5 when his father died, he was still 12 in the Gaiden and it said his father died 5 years before hand. So Kakashi was 7 soon to be 8 and already a Chunin, either way yeah he's been through some shit too but Nagato was 6-7 when both of his parents were murdered right in front of him, 10-13 when Jiraiya up and left never to see them again and 18 when Yahiko died. He lost everybody close to him except for Konan, Kakashi ended up with Gai still by his side so in that aspect they were similar. Both of them have been through hell though, that's for sure. Itachi had to kill his clan and make his brother who he loved hate him. Itachi saw blood and death and war first hand at the age of 4 because Fugaku for some damn reason thought it would be a good idea to show a toddler that, and that moment traumatized Itachi into maturing further than he should be at that age, Itachi later loses one of his best friends that he didn't know he regarded as a best friend until he was already dead and in the process awoke his Sharingan, he then saw his own older brother figure rip his eye out and give it to him, then commit suicide forcing Itachi to awaken his Mangekyo, and to save the village and his brother he did what he did to the clan all the while remaining loyal to the village and protecting it from the outside in his own way. I think its clear that those 3 had it horrible. Gaara is another who had it horrible, his own father made it that way for him. Nagato was a child who lost everything first, then gained 3 people in his life and one of them left after a few years never to be seen or heard from again, and one dies a few years down the road leaving Nagato with 1 person and 1 friend only, mind you his group Akatsuki(the good version he and Yahiko made) was all dead too. Nagato and Konan were left with nothing but themselves, not even friends. Kakashi had Asuma, Kurenai, Gai etc. Itachi had literally nobody after he killed his clan, and Nagato had only 1 person left. So all in all they all lost a lot, and been through hell but Nagato and Itachi had it worse until death while Kakashi continued to gain more people over the years(Team 7 etc)At the end Nagato managed to be a friend of Naruto's but that was literally at the end of his life.
Timeline's all messed up, I remember seeing Kakashi lose his father before that. But Seel's timeline is the most accurate we got lol.
Keep in mind, Gai wasn't a close person to Kakashi when Kakashi lost everyone. The only reason Gai became close to Kakashi is because Gai knew Kakashi was in a bad way and chose to take it upon himself to be by his side, until Kakashi appreciated him.
The only real reason Kakashi eventually found new comrades and friends was really because he developed as a character. Every other character who had been through similar trauma had gone bad and off the deep end, but Kakashi just endured. A true Shinobi.
The same can almost be said for Itachi. I'm not saying he's almost a true Shinobi, he definitely was a true Shinobi through and through. But Itachi's hardships were, for the most part, under his control. He knew he had to kill his clan, so he did it. That's a terrible thing for him to go through, but it's different than if, say, someone else killed his clan. Itachi knew who he was and what he had to do, and took it on, even knowing he'd be in solitude. He chose that. This is because of a series-long flaw of Itachi's that made him believe he could do everything himself. (This isn't me bashing him, it's good for characters to have some sort of flaw lol)
Whereas Kakashi had pretty much the worse case scenario on every side hit him at a young age and just kept enduring.
Yes, Itachi witnessed his best friend and brother figure ripp his eye out and kill himself.
But Kakashi witnessed his closest friend kill himself to save Kakashi, and his other closest friend kill herself on Kakashi's own Chidori, which was only complete because Obito gave Kakashi the Sharingan.
The people who went through the most pain in the series all turned evil at some point, only to be turned by Naruto (Nagato, Gaara, Sasuke, Neji). The only ones who didn't are Itachi and Kakashi.
Only difference is, Itachi chose his path for the greater good, and knew that he had to do bad things for the greater good.
Whereas Kakashi was simply dealt bad hand after bad hand. Over and over.
I like your analysis of Kakashi's pain, he definitely endured a lot of his suffering without him letting it change his motivations and Shinobi way. Although, I'm not sure if I agree with the use of the term "evil", at least in terms of Nagato's situation. I'd consider Nagato's motivation a grey area, it's neither evil nor "good" if that makes sense. Nagato just saw his plan as the necessary thing to do in order to achieve true peace, and he wasn't doing anything new per say, but just giving the larger villages a taste of their own medicine; make them feel the same pain and go through the same conditions of war so they understand pain.
To the Hidden Leaf, Pain was the "evil" formidable leader of the Akatsuki who sought to destroy their community and peace. But to the Hidden Rain, "Lord" Pain was their hero that rose their village out of its utter abysmal conditions whilst Hanzo was in charge. Their villagers and ninja idolised Pain as a God that they respected with both fear and admiration, the ninja that Jiraiya came across completely supported Pain. Whether someone is considered evil or not, it's all a matter of perspective.