Don't you love rabbit holes? They're the very reason this blog exists! (I was down one earlier today all about dimethicone copolyols, which I can't wait to share shortly! And I'm currently down one about surfactants, thanks to this question, Doris! Woo!) I love your curiosity, so let's take a look at your question in greater detail.
The short answer is yes, it can be used as the only surfactant. But you know I can't leave it at that, right?
Cocamidopropyl betaine is what's called a secondary surfactant, meaning we generally don't use it alone. It offers great foam stabilizing and a reduction in the irritant level of the anionic surfactants. It offers a great flash foam and some humectant properties. The version I have is very thin, like water, 30% active, and a pH of 5 to 7.
Why do we call cocamidopropyl betaine a "secondary surfactant"?
Primary surfactants are generally anionic, or negatively charged. Cocamidopropyl betaine is amphoteric, meaning it has a positive functional group and a negative functional group, depending on the pH. In an acidic product, like a shampoo or body wash, it'll have a positive charge and be more conditioning.
Secondary surfactants like the betaines have strong, syngergistic interactions with anionic surfactants, like sodium laureth sulfate or C14-16 olefin sulfonates, like increasing "mildness, foaming, and viscosity properties of the formulation". (1) They aren't used for all of the cleansing and foaming in a product: They're used to boost those other properties. (2)
And therein lies the problem: It's not that cocamidopropyl betaine can't clean our skin or hair well, it's just that it works better when combined with something else. It could be a decent shampoo or skin cleanser, but it's not the most efficient use of this surfactant.
If we go back to your question and refer to the ingredient list - water, glycerin, cocamidopropyl betaine, sodium chloride, bamboo extract. fragrance, citric acid, neem infusion - this doesn't look right. The first ingredient is water, the second glycerin, and the third cocamidopropyl betaine. I find it hard to believe that a product that has more glycerin than surfactant would foam or cleanse properly. (If you look at the ingredient list on the site, you'll see the same ingredients in alphabetical rather than percentage used order.) So I'm going to assume that there's something out of order here.
As an aside, I have no idea why they would be including citric acid in this product when the pH is almost perfect, and why they aren't adding a preservative. I'm so glad to see you're using one!
the conditioner you liked and saw this ingredient list: cetearyl alcohol, stearic acids, cascarilla bark, cocamide mea, infused oil, and neem. That's not an ingredient list for a conditioner. There's no water, for a start. When I zoomed in on the ingredient list in the picture, it was clear there were many other ingredients in the mix, with purified water at the top of the list, so the ingredient list isn't complete.
In light of this finding, I think my longer answer to the first question is this - I don't think this ingredient list is complete and I don't think cocamidopropyl bettaine is the only surfactant in this product. Based on the inclusion of citric acid, I wonder if the other surfactant is something with a higher pH, like decyl glucoside, for instance. (Although after you read the next section, you'll see why this might not be so...)
A mini-rant: I hate that companies are getting away with posting incomplete lists, lists in alphabetical rather than descending order, or lists without INCI names. (Lush is terrible for this, and you'll see more ranting about this in the coming weeks!) This list is definitely one without INCI names, as evidenced by saying "kosher vegetable glycerine", which isn't the official name.
Can cocamidopropyl betaine be thickened by salt? Funny thing is that I can't find any information on this. I've been searching for quite a while, and because cocamidopropyl betaine is a thickener itself, I can't find anything about thickening it with salt. I tried it myself, and saw no thickening with salt, although there are so many reasons that might be, so I can't say for sure either way. I'm sorry I can't be more helpful here.
(1) Page 46, Handbook of Detergents
(2) Page 166, Handbook of Detergents