single word requests - Term for someone who keeps rejecting ideas without specifying what they want

There isn't a single word term for how a person with a hidden agenda behaves. But your description was certainly informative and clear. We are all aware of how difficult it is to deal with people with hidden agendas.

They are difficult, confusing, frustrating, and at times quite discouraging. But so are a lot of other people for other reasons. Their motives may be complex, as well. So it cannot be that one word is set aside for this type or person.

However, you can describe the behavior. Off-putting. Discouraging. Confusing.

This kind of behavior is passive aggressive "often overtly ambivalent, wavering indecisively from one course of action to its opposite" but indirectly pushing for what they desire. Sometimes in the church, this is the only acceptable means of asserting oneself because of the self-denial that is expected. In these cases, the people are behaving passive-aggressively, not displaying a passive aggressive personality disorder.