Short answer; no it isn't. No 1 rule of advice with Medieval fiction, politics is one of their worst areas.

Disclaimer: I am more of an England person, so I don't know as much about the other countries, but similar circumstances apply and most fictional depictions in English language are English or English-inspired fantasy.

There was never a point were there was no process for execution. Even when kings were holding court in the full curia regis style the people who would become judges, law clerks, were still their give formality to the process and witnesses and other evidence was always a thing. A finalis concordia was a sworn agreement in Anglo-Saxon England between disputing parties in a royal court, a formal system of settlement (Doris. M. Stenton, 1969, 2019 edition) . While the power to administer justice at their own will was a thing early on for many kings, and many retained it, even then it was done with something approximating a trial. When James VI acceded as James VI, "cutpurses and other criminals" (Calendar of State Papers Relating to English Affairs in the Archives of Venice, Volume 10, May 15th), presumably also those guilty of violent crimes; were brought before him by those eager to gain his favour, but nonetheless they were judged on evidence as James was a stickler for what he saw as rationality which did include procedure and evidence. And even so this was discouraged once he entered the governorship proper because the English were protective of their traditions of court law. So while this example shows something approximating arbitrary justice that survived the end of the Medieval period, it was not completely arbitrary, nor common or popular practise. Law courts of some sort where preferred.

As John Phillips Dawson (1960, 2007 edition) says, the first judged courts we might recognise a little developed during the Early Medieval period. There is the famous example of Henry II of England sending his judges throughout the land, which was one of the key steps to developing (somewhat) independent law courts. Law courts in Medieval Europe were not expected to neutral or clearly independent, but they operate with some distinction from the Crown and could rule against what they wanted, or stay judgement instead of sentencing if they were unsure. Ecclesiastical courts were also existent and they were almost completely independent in Medieval Europe, which is why Henry II fought with the church with the whole Thomas A'Beckett thing, but I digress. Most cases for treason, which the monarch is most likely to be involved in personally for execution cases, were in fact handled by courts with judges. They might involve royal councillors, and the judges were appointed by the Crown, but it was not the king simply commanding it without process. Indeed, in normal circumstances most of what the monarch did was formally written and signed with one or multiple of official Seals; the Signet, Privy Seal and Great Seal in England. In the 15th century, at the end of the Medieval period, it was standard in England to produce copies stamped with each of the full Course of the Seals for most royal grants and decrees.