I have seen this before outside of the US though many will ask you verbally to hold off applying to other agency ads and jobs whilst they work their magic vs having you sign an agreement.
They do this as it doesn’t look great on you if you apply for a job several times via different agencies or a duplicate application from you after the agency themself may have been in direct discussions with their contacts.
Some agencies advertise jobs that aren’t there to lure you into giving them your cv and once they get a hold of if go fishing with it to see which companies bite before actually engaging you. It appears this agency is trying to avoid you accidentally CV spamming companies and looking desperate.
I would suggest verbally agreeing but giving the agency a time period to deliver (unless it’s with a recruiter you know well) after which you tell them that you will part ways. I would also suggest that you arrange for them to give you a plan of what they are going to do for you and arrange how often they will update you with progress.
You are dollar/the product to them and they won’t waste time on finding you an role unless they believe they can make commission, but that doesn’t stop you from controlling your information and how the process goes.