I know several instructors that don't have lesson horses. I have a few myself, mainly for beginner lessons but my students that are beyond those stages all trailer in with their own horses. I don't have certifications (no need) but have a lifetime of experience riding, training & showing successfully as well as a list of students that have also been successful. I wouldn't put too much weight on the certification (with the exception of therapeutic programs) as long as the experience and references are of note.
From the way she sounds though, she didn't come across very professional and I would certainly look elsewhere. Perhaps a start would be to go to a show in the disciplines you are interested in, pay attention to the quality riders and find out where/who they ride with. A lot can be garnered by simply watching and keeping an open ear.
Life is like a camera. Focus on what's important, Capture the good times, Develop from the negatives and if things don't work out, Take another shot.