united-states
In the United States, there are many orders that a court can enter which are generally not appealable since the case is not fully concluded, but that a trial court can certify as eligible for an immediate appeal.
Sometimes this is done because the order concludes the case for some parties without influencing the parties who remain in the case, so there is no good reason to delay the appellate process for the parties who remain in the case.
At other times, the court recognizes that its decision was based upon a close question of law that could have gone either way in the hands of another reasonable judge, and that the decision will profoundly impact the future course of the case causing lots of wasted time for everyone if the issue is reversed on appeal. In this case, the court may prefer to go forward on a more firm foundation to prevent that possibility.
In some other contexts in U.S. law (such as prisoner's petitions), closer to what the question is probably getting at, it was conceived as a way of screening frivolous appeals in cases often brought by people without lawyers. This kind of requirement for permission is somewhat controversial because it presents the kind of tension between an appeal being available when a trial court gets it wrong and the fear that the trial court might try to procedurally wrongfully cover up its own mistakes, that the question suggests.
In any case, it is not just a relic of postal correspondence and travel being more necessary and more costly. Indeed, in many cases (especially in England and Wales, but also in many U.S. contexts), the distance from the court handling the case at the trial level and the court handling the appeal isn't far at all (sometimes the respective courts are even in the same building or within a few blocks of each other).
Also, postal service is slower and less frequent than it was when the practice was first established. At one point, mail would be delivered several times over the course of the day. There are old contract formation cases studied in law school with several rounds letters back and forth exchanged via the postal system in the same day in urban areas.
If anything, it represents a somewhat less modern sense of a judge's role and what the appeal process is about, rather than anything technological.