The basic 20th century version of the Robin Hood story is different from the earliest version.
In the earlier versions Robin Hood is a yeoman turned outlaw who robs people and befriends a knight. The King at the time is Edward, not Richard. The term “Robin Hood” was used to identify an unnamed outlaw - kind of like John Doe - and was in use at least a century prior to the stories. There were a series of robberies associated with a Robin Hood in the Yorkshire region during the 13th century. There was also a historical outlaw who lived in Sherwood forest that matched some of the aspects of the story. In all cases, Robin Hood was a criminal who was romanticised by the public - possibly due to their dissatisfaction with the status quo.
During Richard the Lion Hearted’s reign, he was most often out of the country fighting. He didn’t even speak English - not uncommon at that time. He was a bad king, but excellent soldier. He sold lands and titles to the highest bidder, with very little care for the country. He remarked that he would have sold London had he found a buyer. As for government and taxation, that had been structured by his father, Henry II, and Richard didn’t do anything to change it. Taxation, even in the middle ages, was nothing like it is today. The Saladin Tax, which was collected for the Crusade Richard and King Phillip had pledged to undertake, was around 25%, collected over many years. The ransom for Richard demanded by the Holy Roman Emperor was of equal amount, and the collection was organized by his mother. These funds weren’t just collected from England, as much of Richard’s most valuable estates were on the continent. The total amount was around 2-3 times the annual income of England, and the family pawned the crown jewels to cover a large portion.
Richard left two regents in charge. John was not one of them, which was the cause of his displeasure. While John did plot to keep Richard away from England, along with Phillip who had offered the Emperor a large sum to keep Richard, he acquired his funds from his own lands. The association to Robin Hood from Nottinghamshire and Prince John simply doesn’t match up with the historical facts. This relationship is a modern fictional addition to the classic tale.
King John was so bad of a king that he actually managed to contribute to the cause of liberty, not that that was his intent. His extreme incompetence led to the signing of the Magna Carta, which pathed the way for future limitations on government. Earlier in his career he had been made the ruler of Ireland, but only stayed 8 months before abandoning it - gee, I wonder why?
So let’s assume that the modern version of Robin Hood was true, and that he was loyal to King Richard even though he siezed the very taxes that paid for the King’s Ransom. He participated in using force to redistribute wealth. He supported a foreign policy of continual warfare. He fought against an incompetent ruler who, although he wasn’t in power, would later go on to sign the Magna Carta and establish the Royal Navy (which would one day enable liberty-minded people to travel to the New World and seek out their own form of liberty from this monarchy). Robin Hood is indeed a villian in either version of the tale.