You're right. It is the saliva. I think the reason more people seem to be allergic to cats than dogs is because they lick themselves more so it probably takes a lower threshold before a reaction happens.
If you do wash a cat, I saw a video about washing cats on YouTube. He said their body temperature is much higher than ours so use water than it much warmer than what would be comfortable for you. It will feel cold to them. Then bundle them up when you get them out to keep them warm. I think the reason they hate water so much is because it usually makes them very cold.