No. Your analysis is incorrect and shows a poor understanding of the events of the latter third of the movie. Spock was the only person that could stop Harrison one on one, they could not transport a large troop down because they were extremely low on power and the confusion from the Vengeance
rendered calling on Starfleet unfeasible. But we also have Spock's revenge and his loss of control to deal with. So we have a compromised Spock going against a superman. Khan had to be stopped and Spock was the only person physically able to do it.
As for people cheering, people cheered when Hitler died, when Saddam Husein died, Bin Laden, etc. The general public does not like oppressors or bad people. But this attitude is covered in Kirk's speech, where he said that we must strive for better, and not let our anger get in the way. But of course, you knew this because you watched the same movie I did, so that makes me wonder if you actually payed any attention.