Although there is no statute of limitations, there remains the right to speedy trial, and intentionally delaying charging (and thus trial) in an effort to compromise the defendant’s (relative) ability to defend himself is, well, disfavored. Patiently (albeit diligently) building a prosecution case is another matter, of course, but lengthy delays between crime-commission and crime-prosecution invariably provoke defense challenges.
As for the “vast resources of the state,” you all need to know just how often that is a complete myth. Many prosecutorial agencies are run on shoe-string budgets, and not uncommonly, public defender offices have considerably more investigative and forsensic resources at their disposal. In some parts, publicly-funded defense counsel are actually paid a good bit more than prosecutors. In California, for example, the attorneys who work for the state agency that represents death row inmates in their post-conviction proceedings make as much as 20% more in salary than their counterparts at the state Attorney General’s office.
None of this, of course, has much to do with the double jeopardy clause–which is a decidely good provision IMHO, because it furthers one of the most important attributes of any rational legal system: finality.