You make a valid point, vanilla neural networks cannot give you more than a point estimate of class confidence. If one wanted to actually gain an idea of variance, you need a framework that allows such a mechanism.
A popular methodology to this is Bayesian modeling. In other words given some data, $\Omega$, you want to create some form of descriminative model $p(y|x;\theta)$ where $\theta$ denotes the parameters that are random variables. This is unline NN's where they also learn some descriminative model $p(y|x;\theta)$ where $\theta$ are fixed. This difference is key to your goal, because if $\theta$ is fixed, $Var(Y|X) = 0$ since any time you put in the same $X$ into the network, itll always come out the same. On the other hand a bayesian model will have a variance, $Var(Y|X) = E[(Y - E[Y|X])^2 | X]$ and $(Y - E[Y|X])^2 | X$ is no longer gauranteed to be 0 and can be empirically measured through some MC method (or analytically based on your model).
Note that your idea is on the right track, that the more data points you have, the more confident your estimates will be, but the only issue is that just considering $\frac{P}{(P+N)}$ in a neural network will be just a heuristic and is difficult to quantify its association to the variance.
I think this may fit your fancy: high level overview of bayesian network blog post. This is a high level overview of using a neural netowrk paradigm but including uncertainty in the weights. Theres tons of tricks out there to make it so you can still even train these with gradient descent and such.
