The feds used to allow you to appeal your sentence forever. I mean if there are problems with a sentence, the government should want to fix it, right?

But then they decided it was too expensive giving convicts access to the courts. So they changed it to I think 7 days. But they decided that was too short.

So the compromise between forever and 7 days? 14 days. If you don't appeal within 14 days you can only appeal on a very narrow scope. Now realize, those 14 days after sentencing you are being transferred from a federal detention center (fed jail) to a prison, either via con-air or prison bus, cross country, staying in various country jails with minimal access to your lawyer or a legal library if you can't afford a lawyer.

The American Justice System is designed to appear like a justice system but to in actuality be non-navigable unless you have expensive paid lawyers working for you. It is very much a multi-teared system. Have you ever tried canceling the WSJ? Imagine if every single step of a Justice system was designed to be as frustrating/stiffling/delaying (when every day counts) as the WSJ canceling process. Oh, you are being transported, and you want access to the law library? Well we can only get you that during lunch hours, so chose if you want to eat. And oh yeah sorry that the morning transfer to the bus was messed up and you happened to miss breakfast. Sure you want to skip lunch? We might ship you again any time and you might miss dinner if we do.

Also, certainly for state cases, a lot of appeal routes are not available unless you are actively in prison. Post-conviction relief, and federal habeas corpus are basically only available while you are locked in prison. If you do all your time in pre-trial detention, or your sentence is too short to fully complete your appeal then your conviction is stuck forever, even if you have meritorious claims. For instance, if your lawyer was drunk, high or not a real lawyer, there is no way to appeal that after you're released, you just have to live with the conviction for the rest of your life and all the collateral reduction in civil rights that comes with that until you die.