There's no John II, Richard IV, or Henry IX, for example. Given how common the name John is, it seems strange that there was never a second King John. Is this more because he was the last king from the House of Anjou, or because he was such a notorious ruler? If the House of Anjou had continued on for a few more generations, might there have been a John II?
The same seems true of Richard III. Notorious monarch, but also the last king of the House of York/the Plantagenet line. Henry VIII? Infamous, and the last of the Tudor line to produce heirs.
Maybe it's just a coincidence, or maybe I'm half answering my own question. It just seems strange that in the eight hundred years since his reign, not a single monarch has said, "Hey, John's a nice name. I'm going to name my firstborn son John." Henry, Richard, and John are still acceptable (and common) names for everyone else, after all.
In more recent times, monarchs choose a "reign name" which is often, but not always, their given first name. King George VI, for example, was born Albert Frederick Arthur George. Edward VII was born Albert Edward. Victoria was born Alexandrina Victoria. So, the parents aren't necessarily the one choosing or avoiding certain names.
In the case of George VI, he specifically chose his reign name to be the same as his father's to emphasize the stability of the monarchy, which was in serious doubt after his brother's abdication. Edward VII chose to use his middle name rather than Albert, his father's name, so as not to "diminish" his father's name - "Victoria and Albert" having been part of the British public's imagination for so many years.
As to why monarchs do not choose "infamous" names for their children: I don't know of any source which would give that level of insight into their thought processes. Nothing I've read indicates why certain names weren't chosen - only why others were. Most royal children seem to be named after others in the immediate/recent family, which would necessarily lead to certain "cycles" in names.
(Also, recall that Charles and Diana named their second son Henry, and George VI had brothers named Henry and John. Although none are/were firstborn, it wasn't outside the realm of possibility for any of them to become king.)
It will be interesting to see what name Charles goes with... Charles III seems likely, but given the notoriety of Charles I, perhaps he'll choose something different?
As noted, seems that reign names are usually named after immediate and recent family. There were three kings in a row all named George, then there was a long period of time before that name reappeared. Same with William... After the Conqueror and his son, that name wasn't reused for centuries, appearing again in the very late 17th century.