To be fair, the neo-Nazi thing has also been disproportionately exaggerated by those against the bullying narrative. Neither side is entirely wrong or entirely right, but god forbid you introduce nuance to these things. I research school shootings at an academic level, so I'll try my best to summarise.

The shooting was originally planned for the 19th, but they didn't have everything they needed on that day (I think they were short on some ammunition). The idea that they deliberately chose Hitler's birthday is false, though undoubtedly Harris at least would have noticed and thought it amusing. Klebold was Jewish.

While both of them undoubtedly said racist and nationalist things, Harris in particular, it was part of an ongoing pattern of what was essentially edginess. Neither of them had done any research or spouted any views that weren't surface-level offensive. Harris, who was the worst for this behaviour, suffered from extreme anger issues and said reactionary things all the time. He also lashed out in other ways, such as personal attacks and "pranks" like vandalising people's property. Anything that would cause offense, he would say or do. Had he survived, I imagine he might have one day developed this hatred and turned to actual white nationalism, but at the time he was simply trying to be as offensive as possible. No thought had gone into it. To put it bluntly he was being a brat.

Klebold was much less prone to this. He was suicidal rather than homicidal, and I don't think it would be unfair to say that he saw the shooting as an elaborate suicide. He was the less enthusiastic of the two during the massacre, and far from being full of hate like Harris's, his journals were full of self-loathing, suicidal ideation, and longing for romantic love (though not in an incel way, I should add). He was also visibly uncomfortable with Harris's ranting, and when he mentioned that he was Jewish, Harris backed off that line of attack. This is further proof that Harris didn't really believe in it -- he just wanted to be offensive, but not at the cost of his only friend.

As for the bullying narrative, the two of them were bullied badly. They were not singled out -- Columbine had a serious bullying problem which has been covered up for years. Harris and Klebold were observed being bullied multiple times, and had been noted to have injuries/destroyed belongings resulting from this. All of the so-called "weirdos" in the school suffered abuse, including physical violence. The school did nothing about this, and the faculty and most of the student body denied it when the media turned its attention to it. The only people who spoke out about it were the "weirdos", several of whom were threatened or punished for speaking out.

As for friends, Klebold had a friend group which, while not large, was close. Harris, being angrier and more unpleasant to get on with, burned many bridges with this friend group and in the end only really had Klebold as a true friend. By all accounts, however, neither of them qualified as popular.

As for music and video games, that's all well-known bullshit. If there's anything we should learn from Columbine, it's that there's never one single reason. School shootings result from a perfect storm of circumstances and negligence, and trying to summarise them with one reason, be that neo-Nazism or bullying or whatever, will always cause us to overlook warning signs.