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
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,