It’s because “home” in this formation is an adverb, whereas school is a noun. You can be an adverb- I’m surprised, I’m exhuasted… - but you must be at a noun (or on, or in, or some other preposition).
You mean adjective, right? Adverb describes the verb, like talking “loudly” or “quietly”
This is the adverb form. If it were an adjective, it would be nearer to the noun and not seperated by the verb like in “He stole home plate.” “Home” is modifying the state of being or “am”.
with the addition that most languages - especially romance languages - have irregular verbs and constructions.
e.g. in french you say “I have 30 years” to say you are 30 years old. in English you say “I am 30” to say you are 30 years old. It makes no sense to say you are the number 30 or you have 30 years. But no one really thinks about it.
