American voters care about conviction rate above all else for their prosecutors.
My guess is that it does not.
If "these problems" is the criminals getting away with their crimes? No.
Unfortunately, human nature is such that many peoples' actual priority is venting the intense angry they feel about the real-world crimes.
And for far too many people, harsh punishments inflicted on expendable whipping boys are an all-too-satisfying way of doing that.
It's outright fallacious to want higher convictions (or even just higher arrests), because crimes aren't even being investigated, right? The goal isn't some scalar "number of people convicted" but rather justice in every case. If cases aren't being investigated, that's a specific problem with a specific part of the system, which needs to be addressed specifically. It can't be fixed by tweaking the numbers further down the line, rather that just creates further corruption.
The way I see it, every time the public demands the justice system (prosecutors, police, etc) do their jobs better, the people in the system respond by doing anything but focusing on their jobs. Either raising the false positive rate ("hard on crime"), going on [possibly silent] strike (raising false negative rate), or even rioting to create disorder from whole cloth (then blaming it on the protestors). What is really needed is accountability in general, so that specific wrongs (either way) can be rectified.