Why can't saliva clean "stickiness"?
So I could have sworn I've cleaned stuff off with saliva before, but I wanted to make sure, so I ran a quick impromptu experiment.

I slathered the underside of my left arm with some maple syrup I had lying around. I waited about 10 minutes for it to dry, then I mentally sectioned off my arm into three parts and cleaned each part off in different ways:

The section closest to my hand was cleaned by putting my other hand under running water and wiping the syrup off (basically, getting my right hand wet and making wiping motions across the arm, then returning the hand to the water, until I felt like I was no longer making any progress).

The middle section was left untouched.

The section closest to my elbow was cleaned by licking my hand, then using it to wipe that section down (essentially the same as the water, except I used my mouth as the source of liquid).

After the cleaning, I let everything dry again for 3 minutes. Then I placed some toilet paper over my entire arm and attempted to remove it. Here are the results:

(Sorry for the horrid image quality) 

Unfortunately, I didn't have any skin marking tools on me at the time (I use pretty much only water-soluble inks), so I had to add the lines afterwards.

As we can see, the section of my arm that was cleaned with water and with saliva are pretty much TP free, aside from a small section near my elbow that I think was just me missing some spots. The section cleaned with water was similarly TP-free. The middle section is pretty heavily TP-ed. What you're seeing is what got left behind when I attempted to rip the TP free.

So it seems like saliva actually can clean off stickiness pretty well! This is backed up by the comment that Michiel made, which points out that saliva is 99.5% water.

That being said, running this experiment gave me a possible explanation for why it might seem like it doesn't: it takes a LOT of saliva to do this!

With the water, I was easily able to wash off all of the syrup off of my hand and get a new coating of water on my fingers. Unfortunately, I can't generate that much saliva at once, and so I couldn't wash off the sugar that was already on my fingers that effectively. That made the saliva cleaning take a LOT longer than washing with water, both in terms of how many wipes it took to clean off most of the stuff and how much time it took overall (sadly, I cannot drool at the same rate a bathroom sink outputs water).

There's another possible explanation that springs to mind: when you have syrupy stuff stuck to you, you probably just ate some of the syrupy stuff. That means your mouth probably still has a good bit of sugar in it, which would also make it harder to clean off additional sugar.

Still, based off of this experiment, I feel fairly confident in saying that saliva cleans off sugar stickiness almost as well as regular old water, as long as you have enough of it lying around.

NOTE: I initially tried this with a flavored whiskey, hoping the alcohol would help it dry faster. Terrible mistake! Not only was there not enough sugar in the whiskey to get any sort of real stickiness, most of it basically ran off of my arm right away, so I ended up losing a lot of it. Save your flavored sprits for drinking, use a syrup!