The thing with generalized linear models is that you can choose your self what kind of link function you want. Just like you can choose to include or not to include an x-variable. So the link can be the logit link function or the identity link function (like in LPM) or the probit link of the complementary log-log link. You can even...

(watch our Spunky here is a trigger warning) .... even have the Cauchy distribution (i.e. the Cauchyit link). It is your model and you can choose one that fits to the data.

Also, you can choose estimator. It could be OLS or it could be ML. Even in the LPM-model you can use the ML (just choose the identity link function) so that the differences in variances is taken into account. ML is iteratively re-weighted least squares.

But it doesn't matter that much if you choose OLS. When p=0.2 the variance would be proportional to 0.2*(1-0.2)=0.16, because it comes from the binomial distribution. And when p=0.5 the variance is 0.5*(1-0.5)=0.25 and that is not so different from 0.16.

Of course there can be a continuous x-variable. But it must only be in an interval (x_min ; x_max) so that it gives predicted values of p between 0.2 and 0.8. I guess that the author intended an x-variable along the whole real line from -infinity to +infinity. Then the p would be outside of the interval 0.2 to 0.8.

But I find this amusing:

....has been told by the federal government to use a linear probability model for this

The government is choosing the model for you! Not what fits to the data.