Sounds like a great deal for that buyer's agent, lol, to get you under contract for anything in that city/area.
The terms of your buyer representation agreement should be negotiable.
I will often sign agreements specific to a property (binds the buyer/agent for that one property only), or something short term & specific such as "legal suited properties in these 1 or 2 neighborhoods for the next 90 days." That keeps the buyer open to working with other agents for different proeperty types/locations etc., and pushes the Realtor to work to find that property for you rather than waiting for it to show up on MLS.