We do not even know the details. Kojima may be as much to blame as Konami.
Of course it's all speculation but I think this most recent news implies, if anything, that Konami viewed Kojima as in their way regarding making the Metal Gear series.
Even if Kojima quit I'm sure that Konami would negotiate some way to continue paying him if he gave his blessing to future titles in the series. Again, speculation, but it seems like there was some disagreement regarding the future of the series. Metal Gear Solid V is probably an expensive prospect for Konami, maybe something on the scale of GTA V financially (MGS4 supposedly cost $70 million dollars to make, MGSV probably costs upwards of that).
From a Konami management point of view I could see how you'd make the choice to get another game out in a relatively short time-frame. "Can you do that Kojima?", "No", "Well then, we'll have to find someone who can".