Because there was a set order of succession. You know, first son, second son ex ex. If bastards have just as much right to the throne, and could potentially be made the heir to the throne, things get really dicey. What's to stop a King from making his oldest illegitimate son heir over his legitimate sons? Whats to stop older bastards from trying to claim the throne from their legitimate younger brothers and sisters? Essentially this runs the large risk of making a Christian version of the Ottoman free-for-all succession.
"Bastards can succeed" does not mean "no set order of succession".
Let's take Henry I, since he had a pack of bastards.
Either A) Robert comes first (oldest son, regardless of legitimacy) or B) William comes first (legitimacy giving someone precedence) with any bastards following after the line of legitimate sons is exhausted.
Its just as straight forward as if you have a pack of sons from multiple sequential marriages. You just spell out who is first and who follows.
Meanwhile, as for coups - again, the same potential problem exists with younger legitimate brothers.
The Ottoman free-for-all succession was a consequence of the absence of any firmly entrenched rule on which son came first - given that, and given a system where "legitimacy" (in the sense that taking power by a coup is not kosher), and you can have bastards as acceptable heirs without any more trouble than OTL.