Aside from a lack of training, it is easy to see why Billy would give up becoming the head of Madison Hotels to become a teacher.
While Billy starts out going to school as an immature idiot, he comes to care for his kid classmates in an older brother kind of way, even saving one of them from embarrassment. It is possible he liked the feeling of looking out for kids and decided he wanted that to be his calling.
He ultimately gets along with kids better than adults because of his immature personality. A career in school fits his personality better than working in a corporate office.
After enduring a salvo of bullying and eventually regretting his own past as a bully, he might have become a teacher because he wants to protect a kid from bullies too.
He's genuinely upset his father paid his way through school, robbing him of a chance to grow up. He might want to prevent other kids from being allowed to slide through school.
When you watch the movie at first glance, Billy might come across as the obnoxious Manchild while Eric is the stern and reasonable one. But once you rewatch it, you see Eric also has the mindset of a bratty child. Only his immaturity is a far more corrosive kind.
At the movie's beginning, Eric teases Billy during dinner to get a rise out of him.
Eric resorts to blackmailing Billy's principal. And when that fails, he throws a monstrous tantrum that puts his secretary in a coma and then tries to feign ignorance of the plan. Eric does asshole things and doesn't take responsibility for them.
Finally, at the decathlon, Eric resorts to gun-toting threats and is so wicked he can't answer a question about business ethics, proving that underneath his corporate persona, Eric has the heart of a selfish and cruel child.
At the beginning of the movie, Billy starts out as an immature asshole who embarrasses his father, but over the course of the movie, he does grow out of it. On top of doing his school work and becoming more protective of his younger classmates, he also apologizes to a kid he bullied in school, stops pulling idiotic teenage pranks, and in an ultimate act of humility, he accepts he's not fit to run a major company. He hands it over to Carl and chooses to go to a college to study teaching on his own terms, proving he's gained a work ethic. Billy becomes a much better person, while Eric remains a petty and murderously self-absorbed cretin.
Billy proving not to be a total idiot is foreshadowed by pegging Eric as a total sleaze. Considering Eric eventually resorted to blackmail and pulling a gun on Veronica, it proves Billy had far more common sense then he was given credit for. Thus, one can assume Billy was never truly stupid but emotionally stunted by a coddling father who gave him everything he wanted and let him slide through life.
Eric pulls a gun out over a question he couldn't answer. He had to have been carrying that around with the intent to use it, and the odds of defending himself didn't look that high. In other words, Eric would shoot up a school just to get what he wants!
Eric has proven not to be just some corporate sleaze but a total bastard willing to resort to blackmail, trickery, and death threats to get what he wants. He also has no understanding of business ethics. If Eric did get away with taking Madison Hotels, how much damage could he have done to the company and its employees?
On a more mundane level, Billy's teachers not only let him slide through school after Mr. Madison paid him off but even let him get away with bullying someone to the point that the person, Danny McGrath, had murder revenge fantasies for a long time. So how many other kids did these corrupt educators hurt?
While it is played for Black Comedy, the fact remains that Danny had a kill list for people he hated, and he was, at least, capable and willing to shoot someone in the ass, albeit in self-defense. Did he kill someone in a fit of rage?
