For every 15 we get calls about, I go and look at 3, and maybe take 1 on trial for 30 days. I call 4-H clubs, local ranches- most people have no trouble finding us! The tough part is communicating to people that we need middle aged, fit horses. A good number of the calls we get are horses approaching 30, who are great for the grandkids to be led around on, not going to work! Educating the public is hard! We spend a LOT of time schooling and conditioning and keeping the horses challenged outside of their regular classes- this has really helped decrease burnout, as well as the feeding schedule they are on (free choice slow feed round bales 24/7). I know this wasn't part of the original question, but I throw it out there since keeping them happy as long as possible is tough :)
I do special groundwork exercises too that encourage a lot of relaxation, which a lot of incoming horses need practice with. I am rarely handed a horse that's perfect for program right off the bat. With the right temperament though, I've been able to get some that would have flunked the initial evaluation safely and happily working in classes. The biggest thing I look for is try, and a desire to be with people. That being said, some of our newbies wouldn't be able to handle ALL of our riders (the ones that randomly scream/kick/exhibit impulsivity). I start out all new horses with our consistent, steady eddy riders who aren't going to throw crazy things their way. Once in program for a few months, and having built up their confidence, they've turned into nice versatile mounts. In the end, if they are making it obvious that the job is the not right fit for them, we find them a better home. I never try to fit a square peg in a round hole, and would never compromise a riders safety by keeping an inappropriate horse. So please don't take this as a "you can turn anything into a therapy horse with enough time and training" :) PM me and I can show you some of the relaxation techniques and exercises we use- my instructor (who developed them) and I presented at the PATH International conference a couple years ago with them, they've helped our horses a lot!