I haven't seen Welcome Back, Kotter in years, but the theme song popped into my head a few days ago. I know that his dreams were his ticket out of there, but that they have turned him around, but do we know why Kotter returned to James Buchanan High School?
He had no choice in the matter. A bad job market
From the beginning of S01E01:
— You're stalling.
— I'm not stalling. I'm not goin'.
— Ugh. Again? Honey, just be happy that you've been assigned anywhere. A lotta new teachers can't find jobs at all.
— Julie, James Buchanan is not anywheya. It's in Bensonhurst, which is in Brooklyn, which is where I spent four degenerate years as a student. Y'know how rough that is? The gangs theyah don' use guns. They insert the bullets manually.
and karma forced him into it.
The principal hadn't changed in the ten years he'd been out (presumably trying and failing at other gigs before settling on teaching) and still remembered and blamed him for a riot in the cafeteria. He took the opportunity to foist off his troublemakers on the new teacher he didn't like anyway.
— You haven't changed a bit, Kottah. You still have streetmouth.
— I have a degree in teaching, Mr Woodman. That's why we're here.
— You wanna teach? Alright, teach. I'm giving you back your own. The special-guidance remedial academics group.
— You're giving me the sweathogs??
— To you whose class launched that revolting name, they're "sweathogs". To me, they're... "remedial... sweathogs."
— Mr Woodman, d'y'really thin—
— Mr Kottah, you'll find that first aid is still in Room 118.
Of course, later, it all turns out to have been for the best.
— Oh no, please no.
— Kottah, like a prayer that was answered, you quit. You said up my nose with a rubbah hose.
— Remove the hose, Mr Woodman. I'm back! You want me out, you gotta fire me.
— I don't want trouble with the Board of Education. Y'know how I hate trouble with the Board of Education.
— I guess you're stuck with me, then, huh?
Gabe Kaplan was a remedial High School student in Brooklyn. He fashioned his sitcom on the concept of a student from his background, returning as a teacher to teach the next generation of remedial students.
So, to answer the question, Kotter was supposedly a founding member of the Sweathogs, and he was returning to the high school he graduated from to teach the next generation of Sweathogs, who he knew he could relate to and therefore unlock their potential.
"Who'da thought they'd meet ya
back there where they'd need ya?"
They needed a teacher who could get through to the kids. But, in reality, they expected to only get (at best) a babysitter to hold class until the sweathogs dropped out. Of course, since Kotter could relate to them, he was able to actually get them to learn.