This could not possibly be a problem anymore?
A statue of limitations serves a few purposes.
1. At some point closure is better than justice, or at least, that's a theory. Better to get a crime on record because the committer of it confesses to it than not. The idea being that after, say, 10 years, the odds that the criminal would get caught have drifted off to as near to 0 as can be.
2. Records eventually get shredded if you wait long enough; eyewitness recollections in particular slowly mutate into conjecture (whilst said eyewitnesses will believe what they remember is accurate, there's plenty of proof out there that given enough time what they remember is not at all accurate and it gets worse over time). This makes it easier to accidentally convict on flawed evidence, and makes it harder for the defendant to defend themselves.
Many acts are ambiguously criminal, until clarified by a test case. Legal decisions are also often made in the context of time, taking into account what is commonly considered offensive, what a person is presumed to know (in establishing mens rea). It's difficult to try some crimes with a modern perspective and take into account what was considered acceptable in the past.
Its existence points towards having a “legal system” rather than a “justice system”.
Correct, it’s called ex post facto.
(+) arguably it isn't, in ECHR juristictions
"No Bill of attainder or ex post facto Law shall be passed."
Civil laws, such as, immigration laws, are not so limited. That's why I encourage permanent resident immigrants (green card holders) to complete the naturalization process ASAP. (Assuming they want to retain the right to stay in the US.)
Article I section 9 doesn't say anything about the scope being limited to criminal law.