If your daughter has a speech delay, you should see an audiologist and have her hearing tested, and also have an evaluation by a Speech therapist. If she has more than one issue, I would switch from a neurlogist to a Developemental Pediatrician. It may take you a long time to get an appointment, but they will call in every area of assessment that she needs, from speech, to Occupational therapy, cognative, neurological, psychological, physical, social, genetic...everything she needs, all rolled into one report for you, so that you do not have to guess. Pediatricians are not all good case managers for developmental issues, you need a specialist to be sure that you are not missing anything. My guess is, your pediatrician checked to see if she had fluid in her ears, which is not the only cause of hearing loss.
Continue with the other scattered appontments until you can get in to see a Developmental Pediatrician, you can start therapy for the things you see now, like speech, or motor issues. Start as much therapy as you can as soon as you can.
For young children, they will probably do an ABR, when they numb her ears and put an electrode in the ear canal to measure brain activity. It is difficult, but not imposible, to get behavioral hearing assessments done at her age, but they may try. This can't be done in a doctors office, only an audiologist would have the means to do these.
You are right to trust your gut on this, get more, rather than less evaluation. Pediatricians are there for your day to day issues, development is a specailty, and Developmental Pediatricans are the ones to see.
Good luck.
M.