What about non-violent offenses, like the mentioned example of someone who trespasses to get out of the cold? I assume you wouldn't advocate for the death penalty in that case. So wouldn't you want them to figure out why that person is destitute and try to fix the problem, if they are going to be in custody anyway?
As to DNA, don't believe everything you see on TV. In real life, DNA evidence is usually a messy, inconclusive  chemical analysis where there is always a chance of a false match.