Fitting a psychometric function when data does not lend itself to a sigmoidal fit
What you are looking for is called Hierarchical, Multi-level or Random-effects model. In your particular case the solution is a hierarchical logistic regression. 

Assume $y_{st} \in \{0,1\}$ is the response of subject $s$ on trial $t$ and $x$ is the dependent variable then a simple hierarchical model that solves your problem is: 

$y_{st}\sim \mathrm{Bernoulli}(\mathrm{logit}(\alpha_s+\beta_s x))$

$\beta_s \sim \mathcal{N}(\mu,\sigma)$

where $\mu$ is the population value of the slope and $\beta_s$ is the subject level estimate. Roughly, $\mu$ is a weighted average of all $\beta_s$ where the weight of each $\beta_s$ is inversely proportional to the variance of the estimate of $\beta_s$. For more details on hierarchical logistic regression and for extensions of the simple model, that I have suggested above, refer to Chapter 14 in Gelman &amp; Hill (2006).


  Fitting a function to this data, the slope should be nearly flat, right? 


No. The slope should be uncertain. Flat slope looks differently, say $(10,0.61), (20,0.59),(30,0.6),(40,0.58),(50,0.6)$. The corresponding estimate of $\beta$ should show wide interval such that you can't conclude that $\beta&gt;0$ or $\beta&lt;0$ or  $\beta=0$ (as you suggested).

How will a hierarchical model handle such uncertain $\beta_s$? This $\beta_s$ will contribute little to the estimate of $\mu$. Instead $\beta_s$ for this particular subject will be pulled towards $\mu$. The hierarchical model will effectively tell you that if your data is inconclusive, it will just assume that the subject is a typical member of the population (that is if $\mu$ had been estimated reliably) and discard the erratic data.

Literature: 
Gelman, A., &amp; Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.