Yeah, I think they just overlooked the need for it in TOS. If private conversations were needed, they were done in the turbo-lift, conference room, sick bay, or personal quarters. So, they may not have even considered it. Even if they did, it was probably axed because it would've required yet another set piece and added to the costs.
We do see in TNG, VOY, and ENT that the ready-room can be an essential place of business, for carrying on interactions in private. I'm certainly glad they were present in those series.