Look at your contract and see if they breached it. Insist that you have access to the applications of those that have passed the initial screening so you can make sure they are not lying to you. Tell them flat out that you do not trust them because of the way they have been handling the situation so far.
Did you sign an exclusivity provision or is there any language in it that would bar you from pursuing renting out the property on your own (placing your own ad and looking on your own, etc)?
If not, I would try to go my own way and find someone to your liking before they do.
If you must continue with them, I would give them a strict, no if ands or buts set of requirements and procedures that must be met before showing the property.
Set a firm income level and firm credit score as well as debit to income ratio. Make sure your policies on pets and smoking are clearly understood. Tell the realtor that the rental price is non-negotiable and to not even entertain the thought to bother you with trying to convince you to take a tenant who will not meet the price because you know they will try that.