He's not a normal person though. The shogun is a guy that's in power of an empire, well whatever's left of it. and has been paranoid the entire time he's in power from what we can tell. He's likely had many people try to assassinate him prior to that incident. Him fearing his son would be pretty standard. How many stories involve royalty fearing being assassinated while on the thrown. Heck, it happens in real life, as I recall lol.
Biba didn't try to kill the entire human race. He let the Kabaneri loose on the capital and the town that guarded it. As I pointed out, he didn't do so with the town where he was introduced. Granted, he still brought them there so that he could further gain influence with them but he didn't let the Kabaneri loose on that town at all. I think his reasons for doing that shit makes sense. He was betrayed by his father twice, he was also betrayed by the people under the Shogun, and he had to survive the Kabaneri all by himself. I think think they could have shown more of it, for sure. However, that would just prolong the show and Biba isn't really meant to be a villain we understand deeply or sympathize with. The guy ain't normal lol
Edit: He also let the Kabane into the towns to create the Black Smoke from the Nues. The first town had the giant gate he had to destroy and it looks like he wanted to destroy the Shogun's Palace in the capital with them.
He was King Lear!