The main issue I have with the "desire to protect" construction is that there's many anime girls that I find moe that, well, simply don't need protection.
Does Yui Hirasawa need protection? What does she need to be protected from? Nodoka? Her sister?
I'd only feel a desire to protect someone if I felt that someone was in actual danger, or likely to be subjected to significant harm.
With the above in mind, I have to admit that for most of the times that I use "moe", "kawaii" would work just as well.