The statistical model is a Bernoulli distribution where the expected value p is parameterized by the logistic sigmoid function applied to a linear model of the input variables.

The choice of the use of the sigmoid function does not stem from applying MLE to the model: instead it is an priori and arbitrary modeling decision that has been done even before starting to think about the estimation of the parameters of the model.

The intuitive justification with respect to the choice of this link function given in this blog post seems quite standard to me. This text-book chapter gives a similar intuition: http://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch12.pdf (section 12.2).

Also in practice we don't use pure MLE but l2-penalized (or l1-penalized) MLE to fit a logistic regression model as otherwise it might be very sensitive to noisy data if the number of features very large (compared to the number of training samples).