Sin's clearly alive if it's killing people.
That's not a fact. That's not what Paul said, so it's not a fact. Translating "through" into "aroused by" in the sense you're asserting forces the Law to become a cause -- when it's clearly not.
What's really regrettable about this translation is that 7:4 actually uses a word normally translated "roused".
Get a wooden translation from Greek, or an interlinear. You'll see how much liberty this is taking with Paul's statement. Paul attributes this whole action to sin, working through the Law just like it works through everything: corrupting our use of it. Sin was working through the law to bear deadly fruit. Now'f you want to listen to someone who translated interpret for ya, that's your business. But don't attribute it to Paul.
Sin's the problem. The Law is words on paper, it's not something that's going to express power over sin, but will simply declare its penalty.
And what's sin do? "Oh, look, a great description of sins we could do, right here in the Law!"