I'm actually not aware of any place in the US that routinely administers what could be considered a speedy trial. It's a shame, really, but perhaps an inevitable side-effect of the ever-increasing burden on the part of the prosecution. Pre-Warren Court, it was normal for criminal cases to take days between arrest and trial.
> Our jails are full because we have unnecessarily long sentences and a prison system that is meant to be punitive instead of rehabilitative
I make the point in a sibling comment that the US actually has lower incarceration rates compared to its peer nations, relative to their homicide rates. Our jails are full because we have a lot more crime. We can agree to disagree on the "root causes" thereof.
(Also possible, but I suspect less likely, is that our homicide rate is higher but not in proportion to the number of murderers; maybe we have a bigger problem with serial killers than other countries, and thus have fewer murderers to incarcerate relative to the homicide rate.)