Prior to Edward III the last four kings (John, Henry III, Edward I and Edward II) had been succeeded by their eldest son and since the Norman Conquest there had only been one peaceful inheritance that wasn't by an eldest son (and that was Henry II getting the throne in lieu of Stephen's son!) So when the Black Prince died with issue it was unclear what should happen when the king died. Had the crown been any ordinary title it would pass to the Black Prince's children, but the precedent of John's rise to the throne in 1199 implied that a son of a king should be favoured over a grandson of a king, and so John of Gaunt (Edward III's eldest surviving son) should be heir.
Edward III's entailment of the throne in 1376 stated that the crown would be inherited by male-only primogeniture (the future Richard II, then Gaunt and his son the future Henry IV, then the future Duke of York and his sons). However, the choice of this rule of succession appears to have been an act of politics, so it cannot be assumed that the same rule would have been chosen if it would have resulted in a different heir.
John of Gaunt's massive unpopularity at all levels would have led to trouble if he was heir. He had pretty much governed the country since 1374 (Edward III and the Black Prince being ill) and was blamed for high taxes, corruption and the loss of land in France. On the other hand as the greatest landowner in the country having John of Gaunt dissatisfied with the inheritance would not be a recipe for stability.
My view is that if Richard of Bordeaux had been instead a girl then England would have returned to its old custom of disputed successions. I think the most likely result would be John of Gaunt being named heir, but having to fight against an anyone-but-Gaunt faction led by Edmund Mortimer that might be aiming to put on the throne either the Black Prince's daughter (with a suitable husband lined up a la Guildford Dudley) or Edmund's son Roger, who at the time of Edward III's death would have been the only male descendent of either of Edward III's two eldest sons.