Few points about the mess that probably shouldn't have happened.
Edward V in London quicker and not in Richard's power almost certainly makes any coup harder but not impossible - the longer he remains on the throne the safer he is. I am one who believe the coup's motivations were entirely down to misunderstanding and the resentment and power envy of certain individuals rather than any crack in the family relationships of the House of York.
Edward V was not a stranger to his family - although largely based at Ludlow - he saw his parents relatively frequently - his household was large and his closet adults were his Uncle Rivers and his half brother Sir Richard Grey along with the wider members who made up his council who effectively governed Wales on the King's behalf. It should really be thought of as a court in miniature. We know virtually nothing about his relationship with Rivers, who was well rounded and highly regarded which was why he had been chosen as governor. It is impossible to tell whether his reign would have favoured his maternal uncle or half brothers over his paternal uncle and cousins.
It is perfectly reasonable to assume had he reigned his relationships with his family would have changed and developed - and technically if he was like his father then it not unlikely that all of his relatives might have found themselves in or out of favour. In fact any childhood resentment of River's might have meant it was Uncle Anthony who fell at some point in the future.
Key players and alliances.
The Woodville's did not really act as a power block by the early 1480's - the faction if you can call it mainly consists of The Queen, Anthony Earl Rivers (governor of the Prince of Wales - he had a good reputation across Europe), Thomas Grey Marquess of Dorset (the Queen's son by her first marriage - close to the King and a rival to Lord Hastings) and his younger brother Sir Richard Grey. (The Queen's other surviving brother's Edward and Richard Woodville played very little part in events, her other brother was a Bishop but a young one and appears to have been more supportive). In fact the ease with which Richard took control provides evidence for their general weakness as a faction in terms of money, land and support.
In terms of financial gains from Edward IV - only Dorset was a major beneficiary - largely through his marriage to Cecily Bonville and the betrothal of his son to Anne St Leger daughter of the King's eldest sister - the vast Exeter inheritance had been promised to the young couple (the estate had been bestowed on Edward's sister after her seperation and divorce from the Lancastrian Duke of Exeter). River's gains from the King were minimal and based largely on his court position.
Richard of Gloucester, wealthy and up to 1483 very loyal to the King - married with only one legitimate child - effectively ran the North of England and recently the war with Scotland. His relationship with the Queen was on the whole good up to 1483 all the emnity between them post-dates his usurpation and could be argued to be propaganda and justicification - it is quite hard to judge what is truth and what isn't. Suffice it to say his relationship with Rivers and the Queen appears to have been amenable.
If anything Richard's actions can be prescribed to two things - the machinations of Lord Hastings and Buckingham both of whom resented Rivers and Dorset due to their specific regional rivalries. You could add Richard's own ambition and the power vacuum left by Edward's death depending on your view of Richard.
Lord Hastings - oldest friend and courtier of Edward IV - key member of the council. Long-term rivalry in the Midlands with the Grey's for influence, not the best relationship with the Queen on the surface, his relationship with Dorset was poor despite Dorset's wife being his step daughter. He appears to have urged Richard III to act to take control - the only real reason for this appears to have been his fear for his own position - Richard executed him shortly after taking control largely it is assumed because he would have opposed Richard taking the throne.
Duke of Buckingham - despite his Lancastrian background he was a figure at Edward's court and married to the Queen's sister - his resentment had several origins - the fact that despite his dukedom and connections he never gained the influence his ancestors had done particularly in Wales where the control he could have expected had passed to the Prince of Wales' council (effectively Rivers) and other Yorkist peers, he also had a claim to the de Bohun inheritance that had never been settled.
He supported Richard's coup and would be well rewarded but would rebel later in the year.
On Edward's death the council met - it was under no legal obligation to accept any rules Edward IV devised for the government of the realm during his son's minority even had he done so - they agreed to summon the King to London in Rivers' care, they limited the size of his escort (largely due to Hastings intervention) - it appears from some sources the council was in disagreement about certain things but according to some sources this was settled with the intervention of the Queen (how is not known but I suspect it was some mediation between Dorset and Hastings). At no point did the council at this time discuss any question of a regency or protectorate nor did they change the ordinances establishing Rivers' as governor of the Prince of Wales' now King. It seems likely given the age of Edward V that they intended the council to rule in the young king's name at least until his coronation and it may be they intended to declare him of age at his coronation.
River's was slow leaving Wales and took a long route stopping frequently to show off the young King and holding masses for the late King. It does not suggest that he had any concerns about the immediate future. Richard III held a huge service at York for the late King expressing loyalty to the new King and wrote to Rivers arranging to meet him so he could enter London with them. Whatever happened to change things happened in the next few days when Richard and Buckingham finally met up with Rivers he was arrested and they took charge of the King and the rest is as they say history - suddenly Richard is protector (as named by his brother he claims, Richard was under threat of destruction by the Queen's family, and then the king is illegitimate and Richard declares himself King, conveniently overlooking the claims of his other nephew Edward of Warwick).