I should have said 'up to' his sixties, which wouldn't be unreasonable for a well-off male of good physique in the 14th century, assuming he dies of natural causes. You name Henry II, Edward I, and Edward III as being exceptional, but fail to take into account the fact that Richard I died of a gangrenous wound, John died of an illness he most likely wouldn't have contracted had it not been for his war against the barons, Henry III lived to 65, and Edward II was murdered.
I don't think John's death by illness should be discounted because of campaigning. Disease is a serious hazard in this era.
But if you want to look at the whole list of kings descended from Henry II - not counting those dead because of violence:
Richard: N/A
John: 50
Henry III: 65
Edward I: 68
Edward II: N/A
Edward III: 65
Richard II: N/A
Henry IV: 46
Henry V: 35
Richard II: N/A
Edward IV: 41
Edward V: N/A
Richard III: N/A
Average: 53.
50+65+68+65=248
46+35+41=122
248+122 = 370. 370/7=53.
This is better than the French average in the same period (not counting the baby John that may or may not be counted as John I), I think.
It being likely that Edward will act as a warrior king, of course, it is more likely that Edward will die in battle or contract illness while campaigning, but I really do not see it as being as far-fetched as you do that Edward could live to 60, especially given that two of his brothers lived to 58 and 61 (also Lionel of Antwerp who brings the average down for Edward III's adult sons may have been poisoned - I do not buy into this theory myself, but it is worth paying lip service to).
I've never heard that theory (but what I know of Lionel of Antwerp is minimal, so any good sources would be welcome), but even if we take it seriously the average for the three sons that weren't murdered is 55.
That's rather short of Henry III, Edward I, and Edward III - and around the average for the Plantagents in general.
I wouldn't object very loudly in any given timeline for Edward IV to reach his late sixties - but when weighing the odds, the odds favor falling short of twenty years on the throne, more like half that (which would make him 56).
Still long enough for Richard II to come to the throne as an adult (19), but a very young king nonetheless.