It just seems to me he went to so, so much more trouble setting himself against the Church, divorcing his wife, making myriad enemies in Europe etc; when FitzRoy could've been his last ditch heir to the throne if Queen Catherine proved incapable of bearing a son. Henry even acknowledged him as his son. He could've told Anne Boleyn to be happy as his recognized mistress or kick rocks right? With the Tudor dynasty being so much more stable than the monarchy was in the Plantagenet days, why wasn't this considered more of an option for Henry?
While other answers and perspectives are welcome, you might be interested in the ansuwers by /u/itsakidsbooksantiago and /u/PlinytheHipster in Why didn't Henry VIII legitimize Henry Fitzroy, his bastard by Bessie Blount instead of going through all that hoopla with Catherine and Anne?
Well, for one the Tudors were much less stable than the Plantagenets. They would have been essentially seen as a continuation of the Plantagenets, the separateness of dynasties was usually not mentioned in favour of a more continuous view, but Henry VII was an unlikely claimant; occurring because there were few others left and he had the support of both the surviving Lancastrians and disaffected Yorkists including Dowager Queen Elizabeth Woodville. At the time of Henry VIII there were still many who questioned whether his father was the best remaining claimant, which we can see in the activities of Edward Stafford, 3rd Duke of Buckingham, the Courtenay family and the Pole family who were targeted for being suspected aspirants to the throne. His father claim's came through the legitimised Beauforts, as mentioned in the answers linked, and it was also through his grandmother. His mother was from the Yorkist line, generally considered to have a better claim via primogeniture, but she was also a woman.
To have his known bastard son, even if he was legitimised, as his heir would have furthered the likelihood of plotting and revolts. Every monarch in the Tudor line faced plots and rebellions, having a bastard king would have made that problem worse. It may seem easy to hold the throne from the modern perspective, given both Henry's and all of Henry VIII's children managed to survive but any of them could have been overthrown as Jane Grey was or Henry VI. It was simply too big a risk. His sisters Mary was married to the King of France, can't possibly make him king, then Charles Brandon, a subject born without hereditary title, and Margaret was married to a Scotsman, also untenable. And Henry did not trust the claimants from the Yorkist nobles, who in any case relied on a claim that would cast shade on his and thus couldn't be recognised.