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You’re thinking in terms of location, rather than state-of-being. “I’m home” is your status.
“I’m driving, I am bored, I’m safe, I am away”… None of those sound weird, do they? This, combined with the more technical grammar rules others have commented…
Hi, home! I’m dad.
Solid.
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).