There is a question at the end of Kahn’s Calculations in etale cohomology which seems to be directly related to etale homotopy theory for varieties over finite fields.
Find Weil-etale homotopy type.
To recover Brauer group, try to look at the corresponding picture in Hodge theory (NS, Pic, 19+3 = 22 etc). What would you do in Hodge theory to recover these parts?
nLab page on Etale homotopy