Light had a good idea going. "Destroy all of the worlds criminals, and no one will commit any crime ever again!" However, that has three major flaws.
1) What defines a "criminal?" How far should he go with that? Only kill murderers and big-time thieves? Kill anyone who so much as lies or does something selfish? It's too vague.
2) No matter what you do, there's bound to be a couple of murderers and whatnot left over anyway. People often do things simply because they shouldn't. I guess that wouldn't be too much of a problem, but...
3) Light couldn't live forever. Any way you slice it, he just can't. He could always try and find an heir, but does that ever work? They always have
slightly different views than their mentors.
Ryuzaki basically thought the government was doing everything right. He thought due process and the justice system would find and eliminate all crime eventually. That's not quite true.
I: The police can't catch everyone. It's that simple. Some people can hide, some can do evil in a technically-legal way, and some ARE policemen.
II: The court system isn't perfect. There's always going to be corruption, loopholes, and missed evidence. Some guilty people will stay free, and some innocent folks will be imprisoned. That's fair, right?
III: The government is composed of many people. Not all of them will agree on what's right and wrong. There will be conflict. Light's way is much simpler. There's no disagreement to be had among 1 person.
You know what? As crude as Light's method seemed to be, it's actually a lot more efficient. Only one person, no criminals can escape. The only downside is, eventually he would die. Whoever succeeds him would ruin things.
I bought a cactus. A week later it died. And I got depressed, because I thought, Damn. I am less nurturing than a desert. -Demetri Martin.