I think both points go hand in hand. Vulture wasn't just stealing from big governments, he was stealing powerful technology and selling it to criminals. Criminals who will rob stores, banks, etc. And these weapons were crazy powerful, and could easily cause collateral damage. It's not as simple as the underdog vs the 1%.
For the first point, I think Peter realized Vulture was going after the big stash. He knew villains with Avenger's tech/etc. would be a massive problem. And he knew only he could stop it (well he tried calling Happy but that didn't work out).
One thing I really loved though was even after the Vulture threatened to kill Peter's family and friends, dropped a building on him and left him to die, he still saved Vulture's life at the end of the film.