The easier scenario to think about where statutes of limitation really make sense is in collection of payment through the court system. Suppose you buy something on post-payment terms and then the supplier bumbles around forgetting to bill you for it. At what point should you reasonably be expected to pay the bill? In my state you get 7 years, and I think that's probably pretty generous because it covers the entire tab from when you get the thing to when you start a proceeding in court.

For a robbery that doesn't involve a weapon I think we should generally forgive and forget if it's been long enough. Nobody cared enough to bring action in court for whatever reason, and it would be awful for someone in their 40's to be jailed and brought into court for something that happened in their 20's. At that point if the government fails to prosecute that's on them, and on us for failing to hold them accountable. But 20 years is a long time and people can change over that timespan, so it probably doesn't make sense to hold a grudge for that long.

There are especially egregious crimes that have no statute of limitations like murder and sexual assault, but we might find our society better off for keeping the statute of limitations for injuries that we can recover from.