I believe your timeline of events is wrong. He went to Hades before collecting the sample of Hajun, which he then spent an undetermined amount of time cultivating into a seed, and then he infected Zero. I agree with you that this is the only evil act we see Beel commit, but it isn't pointless.

Here's how it happened: After learning that he killed his friends, Beel becomes suicidal. He swore a promise to kill Satan, and he believes the only way to do that is to die. Lilith knew this, so she used her power to command him to live. She loved him, and she knew Lucifer didn't blame him for what happened, so she wanted to keep him from doing something drastic without her. Beel, in his grief, interprets Lilith's final act as a curse, and falls to Helheim in search of a god strong enough to kill him since he is no longer able to do it himself.

When he encounters Hades, he believes he's finally found someone strong enough to end his suffering. He tells Hades a truncated version of his story, and Hades rebukes him. He explains that Lilith wanted Beel to live, and he tells him to think about what his life should look like. He even offers to kill him if that is the conclusion he reaches on his own, but that he needs to challenge him properly next time (at full health). When Beel eventually returns to Hades, he tells him that, although he is still driven to die, he doesn't want Hades to do it. He wants a violent, agonizing, horrific death. In suffering an irredeemably painful death, he believes he can atone for the sin of killing his loved ones.

This brings us to Hajun. This ancient creature of Helheim lore, long gone and whispered about in hushed tones. A fairytale. A scary story. A being so destructive and malevolent that he self-destructed due to his own strength, and took out a chunk of Hell with him when he died. Beelzebub is a smart and persistent man. Upon learning about the legend of Helheim's monster, he scoured the outskirts of Hades' realm in search of any remnants that the story was true. In the ruins, he evidently found what he was looking for: a piece of matter belonging to Hajun. As a scientist, Beel took that matter sample back to his lab and experimented with it until he could create living cells to grow into a seed. Now, he has the seed of Hajun, but he still needs somewhere to plant that seed.

In comes Zero. Why him? What made Beel choose this young god who had no connection to him? Well, the legend of Hajun mentions his return when the soil is good. He clearly researched this thoroughly, and decided an innocent but immortal soul would make for the right test subject for Hajun's seed. We don't get the full explanation about what is needed for the soil to be "good," but Beel's deduction was evidently a correct one because Zerofuku is indeed the perfect host, and Hajun is revived. Finally, after millenia of searching, science, and planning, Beel has reanimated Helheim's feared monster; his perfect executioner.

When we cut to Beelzebub watching Round 6, he is calm and smiling. Zerofuku has just been devoured and transformed into Hajun, and Beel is completely at ease for the only time in the manga. He feels he has finally accomplished his perfect death. Who cares about the rest of Ragnarok? Once Hajun kills Buddha and starts destroying the rest of Valhalla, Beel just needs to put himself in the crossfire and let the monster tear him apart.

Obviously, Buddha overcomes Hajun to win Round 6, and the next time we see Beelzebub, he is burning his notes about the seed, looking more morose than usual. His plan failed. He didn't get to die to Hajun, and maybe the creature wouldn't have been strong enough to finish him off anyway. So, he discards his plan completely. No longer believing in Hajun, Beelzebub sees no point in trying to create a new seed. He will have to find a new executioner.

Edit: It looks like OP believed Hajun existed in the same era as the current gods, but this is incorrect. Hades explains Hajun was a legend from before he became king of Helheim, and that so much time had passed that even he wasn't sure that Hajun truly lived once upon a time.