@James B. first I congratulate you on taking feedback and working to improve yourself so that you provide a better experience for your applicants.
The thing I try and do is get the applicants to talk. Most folks with a story, want to tell it so if you ask open ended questions you will get their story. Hearing their story allows me to move on and spend my time showing the unit to qualified applicants.
The most important step in the process for me is to request and obtain their permission to ask them a few questions. Once they give this permission, they feel less grilled and "stocked" by my questions.
My philosophy behind my phone contact is to screen out the time wasters. Those that have had evictions, are criminals, and who didn't read the ad and the property is not what they are looking for.
After obtaining their permission, I ask when they need to move. I want to make sure my availability lines up with their needs. I'm usually a month or so out and some people need something right away. Others are looking to move in Sept for my May vacancy.
Next I ask what pets they have. Notice I don't ask "if" they have pets. I assume they do so they don't think we don't take pets and say they don't have pets. I then get info on the pets, dog breeds and ages (no puppies or kittens-don't want them to learn their potty training in our units).
Next is who else would be living with them. This allows them to disclose their BF (who it turns out later has a criminal record).
Next I ask what they are looking for in a place, what features are important. My ads are specific about what are in my units but if they want a dishwasher and my unit doesn't have one. Better to find out on the phone than after they walk into the kitchen.
Next I ask about their rent budget, how much is it? It never ceases to amaze me how many people look for something too expensive and think they will negotiate the rent to a lower amount. I tell them no go on the phone and saves the brain damage of, again showing to someone that can't/won't afford the unit.
Finally I tell them we do a credit, eviction, and criminal background check and ask if there would be any issues with this for either them or their roommates. Again weeds out the felons and other misfits.
If they pass these questions, then they get a showing. If they want to rent the place after seeing it, then all of this information (and much more) is again collected on my paper application which is then independently verified.
All of these questions are asked in a friendly manner with appropriate conversational continuers added in when there is a lull in the tenants response and there seems like more information may be coming.