Convert classifier output for disease to probability using Bayes Method 1
I am given a classifier for some disease that takes as input patient characteristics and has some sensitivity and specificity. 
Hence the classifier is a function c(patient characteristics) = 1 or 0
I can then use Bayes rule to convert:
P(disease | c(patient characteristics) = 1) = P(c(patient characteristics) =1 |dz) P(dz) / P(c(patient characteristics) = 1)
Using classifier sensitivity and specificity to write P(c(patient characteristics) = 1 |dz)/P(c(patient characteristics) = 1).
So, even if all I have is a classifier (which made a decision), I can get some probability estimate.
Method 2
The better approach is to develop an estimator directly for 
P(disease | patient characteristics)
Eg using logistic regression or just never binarizing classifier output in the first place.
Classifiers are heavily criticized in medicine, and I agree that it's a poor choice to make a decision without patient and physician utilities, but why can't we just use the first method to convert the classifiers to probabilities?
 A: Don't overcomplicate your use of probabilistic reasoning by bringing in extraneous considerations.  Regardless of the nature of the particular event (whether it involves patient characteristics or the outcome of a medical test) you can apply Bayes' rule to obtain the posterior probability that the patient has the disease.  For any arbitrary event $\mathcal{A}$ you have:
$$\underbrace{\mathbb{P}(\text{Disease} | \mathcal{A})}_{\text{Posterior}} = \frac{\mathbb{P}(\mathcal{A} | \text{Disease} ) }{\mathbb{P}(\mathcal{A})} \cdot \underbrace{\mathbb{P}(\text{Disease})}_{\text{Prior}}.$$
Every event $\mathcal{A}$ that is statistically related to the event $\text{Disease}$ gives information that updates the probability of the latter according to Bayes' rule.  Whether or not a medical test or a set of patient characteristics constitute better information depends on the data.
Update: Your additional edits to your question make it clear that you are also asking about the difference between conditioning on a vector of patient characteristics $\boldsymbol{x}$, versus a binary classifier $c(\boldsymbol{x})$ that maps the patients into two groups.  The latter entails a loss of information, so you are correct that you are better off modelling based on the unclassified characteristics directly.
