Which distribution of the dependent variable is relevant?
First, there is no reason to think that OLS will do badly on this data. At least none from looking at your dependent variable. However...
Second, Looking at the distribution of your dependent variable is not so helpful. Even regression models that assume Normally distributed errors only expect them to be Normal after conditioning on X. This does not imply that the marginal distribution is Normal, which is what you plot here, I think. So seeing fat tails here is not necessarily a problem.
Models and distributional assumptions
OLS does not describe a model but a procedure (minimise the sum of squared errors). OLS generated coefficient estimates have various properties depending on what you are happy to assume by way of an actual statistical model of the the data generating process. The more you assume, the more you get to say. Any good econometrics text will go into the gory details of this, since econometricians always want to know what they can infer from as few assumptions as possible. But in short, you can go a long way without explicitly assuming they are even conditionally Normal.
Probit regression and the alternatives
Probit does allow you to specify a distribution. Indeed, all generalised linear model, a class which includes homoskedastic linear models for which OLS is the maximum likelihood estimation procedure and probit models allow you to specify the distribution of the dependent variable. You can always do that. And you can often only partially specify it too and get some more general results. The class of generalised linear models provides a good starting point when looking for 'alternatives'.
I don't know if I understand the graph properly because I can't see anything that varies at 0.5 intervals (except the histogram bins, but they are designed to do that sort of thing). So thus far, I see no reason to consider probit models.