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.

OK the quantities of the recipe are wrong, the batter for brioche is pretty loose closer to batter than dough or pastry but you do need to manage your yeast. So here are thoughts for attempt 2

Make sure you only use active yeast( you meed to know the yeast is viable ), when you put the yeast into the other ingredients ideally they will all be at room temp to start with, so get the eggs out and let them in their shells come to room temperature couple of hours lets say.

When you combine the ingredients the bowl shouldn't ideally be colder than the ingredients and you need to be sensitive to the time it takes for the yeast, any yeast, to activate and multiply. The whole reaction is complex, temperature, ratios of sugars age and hydration of yeast so realistically if the mix takes 90 minutes rather than 30 minutes to get going so what, if you were cooking to a schedule you would do it differently.

If, after an hour your not seeing signs of the yeast working you have to suspect the yeast or the conditions and frankly more likely the yeast. to test it take 300ml of luke warm water a tea spoon of sugar or honey and a table spoon full of yeast cover with film wrap and put in a warm place for 30 mins, it it is not making best part of an inch or foamy bubbles then all is not well if it isn't making any decernable foam then put your finger in if the liquid is not cold its the yeast.

I agree that the recipe needs to be looked at for proportions here is the mix i use

660 g flour 8 eggs 16 g salt 70 g sugar 17 g dry yeast 70 ml lukewarm water 440 g butter at room temperature

i let it rise for 90 minutes in a cool rather than warm place knock it back and let it rise a second time over night but i do get a dough not a batter.

I agree with the former comments that too warm and the yeast can die.

have another go with a new batch of ingredients,