I love this story. I've always loved teacher protagonists, but I've never had much luck with writing them. They always seem to be full of themselves.
REST ASSURED, you are not doing this. Harry's coming more off as "I'm jaded, but I've got some pretty common logic to share with you guys". The lectures are written well, so Harry comes across as a high school teacher wanting to challenge his students. I have had teachers like him, so I do believe you have done nothing of the sort.
Although I do have a bit of a quandary with Harry's assessment of Dark/Light. Perhaps it is simply because you have not approached this yet – but – with Harry's current stance, he should just come right out and say, "Good and Evil are one in the same, we only identify them by only human distinctions, the universe has yet to give us something that is actually good or evil,". If you really want to make Harry give up the lectures and move into practical training (which I assumed is what Harry had planned to do – theory and ethics on magic and then teach them how to shit fire), don't leave the main point so subtly hidden.
Moments I particularly liked in this fic so far were: McGonagall's apology (unexpected), Harry's "A stopped watch is always right twice a day," (I'm not sure if you heard this somewhere or you made it up, but it's brilliant), and Harry's passing comment "Tom doesn't bother you when you fly," (again unexpected, one where I hit my head and was like "d'oh!").
Sad to see somebody already got the challenge. I was gonna come up with a good name too.
I await the next chapter with anticipation.