How to approach speech analysis?
I think you are too quick to throw AI at this problem. (As you already noted, keep alternatives to neural networks open: maybe factor analysis or principal component analysis are more suitable, or support vector machines)

First, you have to do more work in the problem domain, to find out that different pathologies may result in different models.

My recommendation would be to start small with one pathology. Use domain knowledge to determine if (and to what extent) the pathology is present in some samples, and how that could be observed from the audio samples, in a limited number of parameters.

For some pathologies, speech recognition would help. You could count differences in the text spoken and the text recognized. The count (and maybe a pattern in those counts) could be an input for the model (not necessary a neural network as you already noted).

Some other pathologies would need audio spectral or temporal observations in the audio samples (unnatural pitch, unnatural pauses). 

To give an example: stuttering.

I would look in the audio samples and spectra for differences in the patterns. E.g. a spectral plot of stuttering vs. regular speech could look like: 

So typically you'd want to detect/measure relative power in some frequency regions as a measure for stuttering and learn your model with such data.

An alternative that might also work is to run a stuttering sample (and a natural sample) through speech recognition, to see if differences can be observed in a countable/measurable way.