Let me think about it is rarely about thinking. It usually means there is a worry that did not get said out loud. Push harder and you get polite silence. Walk away and you lose the thread.
The move is to make it safe to say the real thing. Something like, totally fair, and just so I know I did my job, what is the part you want to sit with? You are not pressing for a yes. You are inviting the actual concern into the room where you can help with it.
Often the answer is small and fixable. A timing question, a budget worry, one more person who needs convincing. You only get to help with it if it gets named.
Wingman catches these moments and offers the calm line, so you respond with curiosity instead of pressure, and the conversation keeps moving on honest terms.