Did any of the European states that developed during the middle ages deserve this title (Holy Roman Empire)? My text for this is Traditions & Encounters by Bentley and Ziegler.
The answer to this is, of course, a matter of opinion. It is also an answer that is very similar to my answer to your question on Voltaire’s characterization of the Holy Roman Empire. I would argue that neither the Holy Roman Empire nor any other European state that developed during the Middle Ages deserved the title “Holy Roman Empire.”
For most of the states that arose in Europe (and are discussed in this chapter of your text), there would be no question of calling them the Holy Roman Empire. These were states that could not possibly claim that title. For example, your text talks about Capetian France and the English monarchy that developed after the Norman Conquest in 1066. It also talks about Christian and Muslim states in Iberia. None of these could claim to be a Holy Roman Empire. None of them had even nominal control over the city of Rome, even if they all controlled some areas that had once been part of the Roman Empire. None of them was officially recognized by the pope. The Muslim states were not even Christian and therefore could surely not be called “holy” in the context of European/Roman Christianity. Thus, most of the states of Europe could not possible lay claim to the title of Holy Roman Empire.
There were some states that could at least have some claim to this title. First, there were Italian states that, at least, were in the heartland of the Roman Empire. But these were very small states and could hardly claim to be empires. Finally, there was the actual Holy Roman Empire. This state had the best claim. Its emperors were crowned by popes. It controlled a large area of what had once been the Roman Empire. Even so, I would argue that it did not deserve to be called the Holy Roman Empire. Its connection with the popes was one of convenience, not one of true religious devotion. It was centered in Germany, not in Rome. It was not an empire because it had only a very loose control of its territories. Thus, it is very hard to argue that any of the states that arose in Europe deserved to be called the Holy Roman Empire.
You could say that the Holy Roman Empire was neither Holy, Roman, nor an Empire. It was corrupt (as wealth and luxuries consumed the church), not Roman (it was Germanic), and it was only an Empire in the sense that it combined a bunch of the German states, and it was a very loose confederation on top of that. It was holy only in the sense that the popes helped elect an emperor...