But wasn't it only due to the accession of Richard II that established rule of primogeniture over proximity of blood(prior of that,the principle of proximity of blood prevailed).As for military victories,wouldn't the nobles want a militarily successful king so that they can get new lands(given the epitome of an ideal king in the middle ages was the king being a highly successful military leader)?And what about if he used his wealth to start bribing nobles and shore up his popularity?Would that help?
This was the first time the two had been at odds with one another in 200 years. Since then, English Kings had claimed the French throne by primogeniture, noble titles had been codified to descend by primogeniture in almost every case and Edward I had given Balliol the Scottish throne based upon his English leaning towards primogeniture. Perhaps these were ad hoc pragmatic actions, but they set a precedent. The Constitutional situation of 1371 was very different from that of 1199. In that period, royalty's status as a Mighty Warlord had diminished (it was needed less, and the nobility weren't keen on having to pay for a few months' worth of dysentery in France or Castile every other year in lieu of making cash from their lands) and the King was now much more a guardian of law and order. We can see this rise in Edward I's whole Quo Warranto shindig. Royalty mattered because it offered stability and profit to the powerful men of the realm through the relationship set down in Magna Carta - the security that their sons would be provided for according to precedence in perpetuity; that their estates would not be requisitioned by a wealthy, powerful King who had conceived a dislike for them, or vice versa; that the high taxation, spoliation and bodily danger of wartime would be curtailed as much as was concomitant with personal honour.
If John took the throne, Richard would always be a source of instability, not only to John himself, but to the Realm. And quite a few powerful people, together with commoners like those who trashed his house in OTL, would be bound by either principle or self-interest to fight for a malleable kid with a strong hereditary right to the throne. If John pre-emptively got rid of Richard, he would be a tyrant, and thus would be fair game to rebels like the Northumberlands IOTL. There's no prospect of a peaceful succession while Richard is alive, so what's the point of either Edward III or anybody else backing a different horse - however many times it's won the Grand National?
Yes, Richard turned out pretty mediocre but there was no way of knowing that at the time.