While it's true that from a gameplay standpoint we have the power to fell Primals and such, the main thing that lets us do this so easily is the Echo - because we're immune to tempering, we're able to take them on with far less danger than is normally associated. There's nothing saying other people don't have the martial prowess to match us, though. In fact the Company of Heroes felled both Titan and Leviathan in the past, albeit with heavy losses.

As for detective work and proving your innocence, it'd be a pretty trivial thing to do if you had a fair trial. Emphasis on fair. The Ul'dahn government is so corrupt now that reality has become a commodity. Lolorito wants you out of the picture, so even if you were afforded a trial there's no way you'd be declared innocent.