Why didn't my brioche rise?
First, try proofing your yeast (mix 110°F water, a little sugar, and a little yeast in a small container; confirm that it foams). It may be that your yeast is (well) past its prime. That's the most likely reason I can think of that your dough didn't rise.

Second reason could be that while mixing, the dough got too hot—heat will kill yeast. Yeast dies somewhere around 130°F, but really you want to keep it much cooler than that. If its getting too hot, consider starting with colder ingredients (e.g., ignore the instruction to warm the milk).

Finally, I'd suggest using recipes that are by weight; flour is much easier to measure by weight. You may be using a different scooping technique than the recipe author.

Also—there is something weird about that recipe. It lists 3½ cups flour in the ingredients, but then only has you add 2½ cups in the steps. I suspect that is a mistake, and explains the batter-like consistency you got.