I enjoy this anime even though characters are sometimes stupid. Especially his father: really no investigation beforehand?

I consider Isaac being a dumbass realistic given that the likelihood he has received significant pushback from anyone is close to zero. There is plenty of lore indicating the aristocracy through greed and stupidity has warped the economy of this world.

The butler did his homework (in other words, the investigation) and reported his findings, but Isaac was having none of that, decided that Elm had cheated his way up the ranks, and the only thing left to do was to unmask the collaborator who knew of Elm’s aristocratic roots.

The plan had been to expose Elm’s failure to uphold his promise and then execute him.