I don't see anyone denying consistency. The question is over how different sorts of rolls can be interpreted.
Can you show me any rules text that supports this?
If you are right, that means that an ordinary orc with AC 7 or so (AD&D) or AC 14 or so (3E) - whom no PC can ever miss on a 19 - can never parry a blow. It also means that no one ever parries better or worse, because their defence is always static.
Here is the definition of "attack roll" from AD&D where Gygax, at least, disagrees with you (DMG p 61):
During a a one-minute mele round many attacks are made, but some are mere feints, while some are blocked or parried. One, or possibly several, have the chance to actually score damage. For such chances, the dice are rolled, and if the "to hit" number is equalled or exceeded, the attack was successful, but otherwise it to was avoided, blocked, parried, or whatever.
Nothing there entails, or even suggests, that a roll of 1 to 3 may not be the best-placed blow ever which is nevertheless parried by a skilled or lucky opponent.