I am not sure what is the reason behind having these in the first place, but one of the features of a private search engine should definitely be the ability to search without cookies. This will have several benefits for users, such as:
There might be other benefits as well.
The second point was phrased rather poorly. Let's try again:
The reason why ask.moe does not work in Lynx for me is the fact that I have disabled cookies. As far as I am aware, it is not possible to whitelist cookies in Lynx. You can either blacklist them (cookies enabled globally and answering "never" to the cookie prompt) or reject all by disabling cookies globally. If you have ever used Lynx on the modern web, you know that answering the cookies prompts gets very annoying very quickly as the vast majority of the web uses a lot of cookies. Thus, to use ask.moe in Lynx, one would have to fight all those prompts, which is just not worth it.
Therefore, ask.moe is currently unusable in sane Lynx configurations.