Henry VII's son is actually not the only almost-King of England to be called Arthur!

To preface this, I recently answered a question about the proliferation of the myth of King Arthur and it's historicity that you can find HERE if you're also interested. I'll be hitting on a few similar sources, so you may be interested in reading that as well.

Arthurian legend was only just becoming the powerful myth that we know it as today, and as I go over in my previous answer on the topic, the historical accuracy of the legend was disputed only shortly after the work of Geoffrey of Monmouth was written, even though it became one of the most widely-distributed written works in Medieval Europe. However, it is important to point out that Arthur is Celtic, and not Norman, Angevin, or Frankish, etc. Geoffrey of Monmouth and those who came after make no secret about the Welsh origin of the story. Geoffrey himself may have been of Welsh descent. One twelfth-century source, the Draco Normannicus, casts King Arthur as being Breton, another area with a Celtic culture and language that came under the control of Henry II of England in the 1160s. Henry married his third surviving (legitimate) son, Geoffrey, to the heiress of Brittany, Constance, in order to secure control of the Duchy. Henry ruled over it until his son came of age.

Geoffrey and Constance ruled Brittany together until Geoffrey's death in 1186. Conflicting sources state that Geoffrey became seriously ill while visiting King Philip II of France, or that he was trampled to death in a tournament in Paris. At the time of his death, Constance was pregnant with their third child, and some months later, gave birth to a son. William of Newburgh, after his lamentation over Geoffrey's death, states:

He had a posthumous son by the only daughter of the earl of Brittany; and when the king [Henry II], his grandfather, had ordered his own name to be given to the child, it was opposed by the Bretons and by solemn acclamation he received in holy baptism the name of Arthur. In such manner the Bretons, who are said long to have expected a fabulous Arthur, now with high hope nurture a real one, according to the opinion of certain prophets expressed in their grand and famous legends of Arthur.

This tells us that, not only were the Angevin Kings (and many aristocratic families, for that matter) more interested in naming their descendants after themselves or other deceased family members, but that Arthur was considered a Breton name. Interestingly, Geoffrey and Constance's other children - both girls - were named after Geoffrey's mother, Eleanor, and paternal grandmother, Matilda. Geoffrey himself had been named after his paternal grandfather (and possibly his paternal uncle) and actually had a paternal half-brother with the same given name. Geoffrey's older brothers were William (who died young), who was named after his maternal grandfather, and several paternal relatives, and Henry the Young King, who was obviously named for his father. These naming conventions were pretty standard for the day. It is only when we get to Henry and Eleanor's fifth son, John, that they seem to have just run out of family names. We can only speculate what Arthur of Brittany might have been called had Geoffrey still been alive at the time of his birth.

Constance ruled Brittany in her son's name. She was married to Ranulf III of Chester by Henry II, but Ranulf was excluded from power and the marriage was eventually annulled. After Henry II's death in 1189, his heir, Richard I, attempted to take custody of young Arthur, and control of Brittany, but was forced to come to an agreement with Constance. On leaving for Crusade, Richard named Arthur, the only son of one of the sons of Henry II, as his heir in the event that he died childless. However, it is unclear if this is Richard's true intention, as he would also name his nephew Otto (son of his sister, Matilda), and younger brother John as his heirs at various times during his reign.

When Richard was killed in 1199, John quickly seized the throne of England. Arthur, with the support of the French King Philip II, rose up against his Uncle and tried to become King of England. However, Arthur was eventually imprisoned by John after laying siege to John's mother, Eleanor of Aquitaine, at Mirebeau. We don't know what happened to Arthur for certain, but all record of him stops in 1203. Stories circulated that he had been mutilated and/or murdered by John in order to prevent any further rebellions in his name. John also had custody of Arthur's eldest sister, Eleanor, the Fair Maid of Brittany, whom he kept imprisoned. John's successor, Henry III maintained Eleanor's imprisonment until her death in 1241. Brittany was instead inherited by Arthur and Eleanor's half-sister, Alix. Consequently, we have not had another King John, either.

That was a bit of a tangent, but the tragedy of Arthur of Brittany's life would certainly leave a poor impression for any monarch wanting to call their heir 'Arthur'. Furthermore, there is the issue that it is a Welsh/Celtic name, which Anglo-Norman rulers would have looked down upon as culturally and racially inferior. Edward I fought substantial campaigns to fully subjugate Wales, and hardly looked upon the Welsh as in any way equal to the English. The reason why it crops up again with Henry VII is that Henry VII was actually raised partially in Wales (and in Brittany), and was of Welsh descent on his father's side. He may not have had the cultural prejudices that his predecessors had. While these certainly may not have been the sole and only reasons why we have not had more Royal Arthurs, I hope it has illuminated some of the problems Medieval Rulers may have had with choosing it as a family name.

Sources

Geoffrey of Monmouth, This History of the Kings of Britain (and much other the other literature cited in my previous post on the topic)

William of Newburgh, Historia Rerum Anglicarum (The specific passage available HERE. This is an old, copyright-free translation, but makes for good general reading)

J.A. Everard, Brittany and the Angevins: Province and Empire 1158-1203 (For the lives of Constance of Brittany and her son Arthur. Everard gives the most comprehensive analysis of 12th Century Brittany, and it's relations with the Angevin Empire.)

John Gillingham, Richard I (For Richard's interesting history with heir-naming)

I don't have a particularly specific source for the prejudice of the Anglo-Normans/Angevins towards the Welsh, though a great deal has been written on the subject. It is usually addressed in any general history of High Medieval England that has sections or chapters dedicated to Wales (such as David Carpenter's The Struggle for Mastery, which is my personal go-to overview for the period). It is also evident in primary source materials. William of Newburgh makes his feelings about 'Britons' clear HERE, and Gerald of Wales, who was probably of Welsh descent, alternatively both laments being treated as lesser due to his Welsh heritage, and derides the Welsh for being what he considered to be backward and provincial, a subject that is treated in-depth in Robert Bartlett's biography of Gerald, and which is interwoven into most of Gerald's works.