Yes I explained it poorly I think - their legitimisation was not in doubt - it was whether or not that meant they had a claim to the throne.
One point the Lancastrian claim to the throne was in part based on conquest (by Henry IV)
He also made a dubious Lancastrian claim of Absolute Primogeniture from Henry III based on the arguement that his ancester Edmund Crouchback was the older brother not the younger of Edward I - a claim no-one believed.
The reason people tolerated his claim was a) he was already in charge and b) that the direct male line (from Edward III) took precedent over a senior claim that was in the female line (the Mortimer and later York claim via Lionel of Clarence). And there was a view that Edward III had explicitly stated that in the event of RIchard II dying without heirs the throne was to pass through the male line - to John of Gaunt and his heirs male.
If you accept that and accept the Beaufort claim to the throne - then the succession in the late 1450s was - Edward Prince of Wales, Henry Beaufort Duke of Somerset, Edmund Beaufort, John Beaufort, Thomas Beaufort, Richard Duke of York (the next in male line succession from Edward III) and his sons.
Henry VI had two half brother's to provide for and in the 1450s the easiest way to do it without offending or attracting criticism was to marry them to heiresses - the richest unattached heiress was Margaret Beaufort - her marriage had nothing to do with any debateable claim to the throne she had but more to do with the fact she would make her husband rich at no cost to the crown.
Until the extinction of most of the obvious lancastrian heirs Henry Tudor was a bit of a nonentity - Edward IV during the 1470s and 1480s never felt particularly threatened by this supposed Lancastrian heir and whilst the BRetons and French used him as something to bait Edward with both were unwilling to back any serious attempt to claim the throne until the House of York imploded on itself and there looked like a chance he could win.
By the 1480s there were no male line descendants of John of Gaunt alive - Henry's mother Margaret was the senior female line descendant if and only if you accept the Beaufort's had a claim to the throne. Margaret's wealth and connections by then (she had remained popular with Edward IV despite her Lancastrian connections and her son's exile) meant Henry's position was raised for beyond what he would have been had Henry VI continued to reign and been survived by children.
Henry himself aware of how poor his claim was claimed the throne by conquest alone.
In any scenario with Henry VI having more children you would so alter the timeline (though i don't believe it would prevent a Yorkist coup at some point) that the Tudor's ultimate victory vanishes completely - as do almost any other twists - Lancastrian's failing to kill Richard of York, Edward IV's brother Rutland living, Edward IV never meeting Elizabeth Woodville etc.
Edmund (if he lived) and Jasper had obvious disadvantages in terms of becoming influential principally the dubious circumstances of their parents relationship. Edmund through his wife's wealth was going to be able to exercise some influence but say Henry VI's reign continues and Edmund's death happens as in OTL then the young Henry's wardship and marriage becomes a bargaining tool for greedy nobles.
In other words he becomes yet another Lancastrian noble but willing to sway with the wind like most of them.
Putting self-interest above dynastic alliances was common and expected.
Few other bits:
The issue in the 1450s I think that makes this more than just a who had the stronger claim is the dearth of obvious heirs - and in the 1450s they were in short supply and certainly weakened Henry VI's position politically (as had his health).
In fact the birth of more children with Margaret of Anjou may have helped stabilise his reign but his reliance on Edmund 2 Duke of Somerset in the early 50s had seriously put York's nose out of joint.
There was also the widespread accussation within the court that Edward Prince of Wales was not the King's son.
There was a significant power vacuum at the late Lancastrian court and a reliance on favourites which had offended many including many of the Neville's, ironically the ones descended from Joan Beaufort, and pushed them into Yorks camp.
York's fight initially had little to do with his claim to the throne (which on paper and by our standards was the stronger one) - it was to do with his and many others distaste for the way the government was being run, the defeat in France (where York had been dismissed as commander), the vast amount of money the crown owed York (for his French and Irish campaigns), and York's view of himself as the senior peer of the realm being largely excluded from court affairs.
Taking the two Somersets - Edmund (who died in 1455) and his son - although financially they were relatively poor for Duke's (as most of the Beaufort's unentailed wealth was held by Henry VII's mother Margaret not her uncle Edmund) - Edmund's court offices made up the bulk of his income - this is one reason they were both so willing to fight to hang on to power against the York party in the 1450s.
As to Holland he was richer, and rather vicious - he was also ironically married to the Duke of York's daughter Anne - despite his marriage he remained firmly in the court party and was widely regarded as having a strong claim to the throne at the time.
Reference Margaret Douglas a century later - her decision to bear her children in England was nothing to do with any concerns over their rights of succession - she was almost entirely educated in England at her uncle's court - her husband Lennox was a Scots exile due to his support for England and his long-running dispute with the Hamiltons.
The issue of foreign birth barring people from the throne is often raised but it is debatable - and on Margaret Tudor's marriage to James IV it was explicit to Henry VII that her children would have succession rights in England irrespective of their presumed Scots birth.
As I understand it, their legitimization was never in question, just their right to the throne. The Papal Bull legitimizing them was never overturned, neither were the original Letters Patent which made them unconditionally able to 'receive, hold, bear and exercise' 'any kind of honours, dignities, preeminences' whatsoever. At very least Henry Tudor would have been in a position to present himself as the rightful heir of the Red Rose and claim the Lancastrian/de Bohun inheritance (the Palatinate of Lancaster, etc). The reception granted to him by the Bretons and French OTL would imply that even abroad his proximity to the royal dignity was well considered. I don't see Somerset or Holland have any any great shot at rallying Lancastrian support opposed to the senior heir by primogeniture and the late King's nephew.
As for the foreigners - wasn't there a whole ordeal with Margaret Douglas crossing the border to give birth on English soil to preserve her children's inheritance rights? I'm not sure. I think the chances of a Portuguese or Iberian infante securing the throne opposing so many native claimants would be decidedly slim, even if they were the senior genealogical claimants.
My point here is not that Henry Tudor was somehow destined to succeed Henry VI. My point is that, in a timeline where Henry VI has a number of children - either a second son to take Edward's place as heir, or a daughter - Edmund and Jasper would likely take the place Lancastrian kinsmen had taken in previous reigns. In a timeline where this ensures another Lancastrian reign or two, Henry Tudor and his heirs would also be, IMHO, the most likely to pacifically inherit upon that main line's extinction.