That first part is not factually accurate. States of limitations are at least as old as Roman law, have been in ours since before America was a free nation, and not just a dodge to reduce workloads. The real reason for them is the latter point you make: witnesses and evidence disappear, people change their stories, etc.
In a common law system, where witness testimony is the paramount kind of evidence, it is an issue that many of the witnesses that the defense might have used might no longer recollect things as well as they once would have. It also becomes harder to challenge witnesses for the prosecution, since the evidence that their testimony is wrong will be harder to come by.
Those concerns get weighed against the severity of the crime, but our system also heavily favors the accused. Better that a thousand guilty men go free than one innocent man be convicted, as they say.
The longer the statute of limitations, the more innocent men will be wrongfully convicted. To the extent the system we want is "no statute of limitations where the accused really is guilty", that is a very difficult system to design. We could try a system, where the statute of limitations can be waived if a judge finds that the new evidence is on its face overwhelming, but that then adds the possibility of judicial capriciousness as a factor in the analysis.
Murder is an understandable exception, given the severity of the crime, but I don't think I'd extend it too much further than that (and I'd take the death penalty off the table after a certain period of time in murder cases).