I've finally figured out how not to miss this quest and the next one (Master of the Isles).
If, like me, you suffer from acute collector's disease and try to capture all those "breadcrumbs" quests, and you get stuck on these two quests, here's a solution that worked for me and will work for you.
As explained above, you need to go through "Pressing the Assault", then "Isle Hopping" (1st and 2nd quest introducing the zones). "Scouring What Remains" was the 3rd and "Master of the Isles" the 4th.
To do this, I strongly advise you to install All The Things to be able to follow the quest sequence of each zone and optimize the time you spend there (some quests are not necessary for the main plot).
Then, take a toon that hasn't done the broken shores quests (the dractyr is good for that, for example. Note that the map that's supposed to be in the class domain is for him in the dalaran citadel, set back to the left of Khadgar and his gang).
The first quest is "Aiding Khadgar", leading you to choose a zone. I chose Azsuna and did the quests up to the quest in the Eye of Azshara's dungeon leading to the first pillar of creation. Bringing it back to dalaran gives you "Pressing the Assault".
I then chose the Val'sharah zone up to the quest for Elune's tear, which takes us back to dalaran and leads to "Isle Hopping".
Then the zone of Highmountain to the quest for Khaz'goroth's hammer, which takes us back to dalaran and leads to "Scouring What Remains".
Finally, choose Stormheim until you complete the Aggramar Aegis quest, which takes you back to dalaran and activates the "Master of the Isles" quest.
The order I tested that worked: Azsuna > Val'sharah > Highmountain > Stormheim.
Be aware that if you go to Suramar it won't work, it won't pop the next quest and you may have to start again on a different character.
Happy farming!