This is a clear example of a well known falacy. “If a little is good, more must be better.” This is something politicians get wrong time and time again.
Most things in life are not a simple linear relationship at all values. The punishment as deterence relationship may well have a nice, well defined, and measureable linear area. But to look at that and then claim, that the relationship will remain stable at the extremes is just silly. Because it can’t. You get into all sorts of non-linear issues.
One, the death penalty is as far as you can go. There is the expression, “might as well be hung for a sheep as hung for a lamb.” This expresses the problem that once a crime has reached a capital level, there is no further punishment possible, and thus no further deterence possible. The falacy of increased use of capital punishment is that it reduces the applicability of the deterence mechanism.
Other non-linear effects come into play. There is an assumption that legal deterence is the only factor in deterring people from comitting a crime. But there is a significant human block to comitting crimes as murder or serious violence. For most humans. If you were desperately poor, at your wits end, hungry and with a family to feed, you would probably be not far away from considering a crime like theft, fraud, or burglary. But the vast majority of people would, even in dire striaghts, simply not even consider violent crime or murder. But a small subset of people, for whatever reason, seem to slip past this sticking point. It may well be reasonable to suggest that they are already acting in a manner in which more normal rules for assessing choices of action are diminshed. They may well find the crime/deterence tradeoff much less strong. So there is a population anomaly. Whilst the crime/deterence relationship can be set at a point that works for the majority of the populace, there is likely a small subset for which it doesn’t work as well, or doesn’t work at the same settings as has been found for the rest.
So you end up with a situation where the crime/deterence function can’t be set optimally, and can’t be used past a certain level of crime.
Sadly, politicians mostly just don’t care. The need to be seen to be tough on crime to the voters will always trump the need to actually tune the system so that crime is optimally suppressed. A populace that is always worried about its safety (even if that fear is mostly misplaced) is vastly easier to manipulate with such simple, tried and true, electoral posturing, than a populace that lives their day feeling secure and happy.