One possible reason could be a dark magic influence on James Potter in the fanfiction. Maybe he's been cursed or possessed, causing him to act out of character and disown Harry.
One reason could be for the sake of novelty. Fanfic writers like to create unique and unexpected scenarios, and this is a very different take on the Potter family. It grabs readers' attention.
Some fans might create it as a form of shock value. They want to stand out by presenting a very different and controversial take on the relationship between James Potter and Harry.
The threat to his friends. James is very loyal to his friends, like Sirius, Remus, and Peter. If they were in danger or if something bad happened to them, it would be understandable for him to cry. For example, if one of them got seriously injured or was in a life - threatening situation, his emotions would be on full display.
The disownment could also change the relationships between other characters. For example, Harry's friends might become more protective of him, or some characters who were once neutral might side with Harry against James. It can also lead to a rift in the wizarding community if James is a well - respected figure. People might take sides, causing political and social unrest within the magical world. This can completely shift the focus of the story from the original plot to one centered around family drama and its consequences.
It might be related to power dynamics. If in the fanfiction, James has a lot of power and misuses it in a way that affects Harry or those around him, Harry would hate him. For example, if James uses his influence to push through unfair rules or to marginalize certain groups that Harry supports. Also, if the fanfiction changes James's personality to be more self - centered and less empathetic, Harry would likely dislike him.
Perhaps James had some hidden magical ability that he was unaware of until that moment. It could be something that was passed down through his family but had never manifested before. This ability could have protected him from Voldemort's killing curse. Or, it's possible that there was a plot by some other characters to save James. There could have been a group of wizards or witches who had been secretly working to keep James alive because they knew his importance in the fight against Voldemort. They might have used some sort of time - turning magic or a protective charm that Voldemort couldn't break through.
One possible reason could be that he was poisoned. In the wizarding world, there are many dark spells and potions that could cause such a reaction. Another reason might be extreme motion sickness if he was on a wild magical ride, like on a particularly unruly broomstick or a flying magical creature.
One reason could be jealousy. Maybe Albus has some special ability or is getting more attention from their parents. Another could be that James is just following in the footsteps of the old school bullies he might have known, thinking it's normal behavior. Or perhaps he sees Albus as too different from what he expects a Potter to be.
It could be an exploration of the darker side of human nature. Writers may use these characters to show how relationships can go horribly wrong, even if it's not in line with the original story.
I'm not sure specifically about 'james disowns harry' fanfics being extremely popular. There are so many fanfics out there that it's hard to single out ones with this exact plot as being super popular. But in the vast world of fanfiction, any plot that involves a major relationship change like this could potentially have a following.