Both rulers died violently at the hands of their own people.
I don't think there is a real parallel between the two rulers in this respect. Richard III was killed by the forces of a contender for the throne - not a spontaneous uprising of his own people.
I do think it's a parallel, although certainly not an exact one, as you point out. (This is actually the kind of discussion I hoped we could get going on this topic.)
However, both Richard and Nicholas were in very precarious situations regarding the stability of their reigns, albeit for different reasons. And they shared the problem that this precariousness put a particular premium on their each stepping carefully. Rulers can make a slew of missteps in settled times, and the prestige of their office and the inherent stability of the underlying machinery of government can carry them through it. In unsettled times, placing more steps adroitly becomes critical.
Richard failed to find a way to reach an accord with competing dynastic claims, something which is difficult but has nevertheless been accomplished by adroit rulers on occasion. (in fact, Henry VII, who seized the throne from Richard, solved the problem with a politic marriage. And Henry's granddaughter Elizabeth solved a dynastic and a boundary dispute with an adroit designation of a successor in the person of James I/VI.) Nicholas managed successively to alienate his nobility, the Church, his ministers, and finally his military commanders. When the dust of those collapsed columns of autocracy settled, Nicholas found that the one column on which he put his foremost and rather mystical trust -- the peasantry -- had been a illusion all along.
Richard's death marked the end of the Middle Ages.
Only if you're in England - most modern historians make a far less precise distinction though the 'early modern' period is generally dated from the fall of Constantinople in 1453. But one could argue for earlier dating in, for example, Italy or Portugal.
True. And I don't mean to suggest that either man brought an end to an era, which is why I said only that they
marked an end. I do think each of their reigns each came at or near the ends of periods throughout which the way the world saw itself had been shifting onto a new plateau. Monarchy as a form of government had been under attack since 1789, with each decade bringing more and more chiseling at its foundations. And, as you point out, the advent of the modern area is somewhat amorphous in date, and varies from place to place. But, by any measure, it coincided generally with Richard's life and reign. Certainly after Richard the reason for and the way by which kings went to war changed, with war -- even dynastic war -- becoming more "corporate" and less personal.
Did Richard cause that shift? No. Did his death coincide with that shift? I think so. From the end of the ancient era (which I'm marking here as the first sack of Rome in 410) up to the time of Richard, at least 8 European kings were killed on the battlefield; after Richard only 2. And both of those deaths (James IV of Scotland in 1513 and Charles XII of Sweden in 1718) were in national wars of conquest or alliance, not personal wars of dynastic ambition. And by Charles' time, a king's taking to the field of battle had become an anomaly, not a commonplace. The purpose and nature of warfare had shifted, with fewer kings taking the field and for different reasons, as the process of consolidating principalities into kingdoms was coming to an end (except most notably in Germany, where even there consolidation under the Hohenzollerns proceeded by a very different means than the personal battles of an earlier age).