A few thoughts:
-The oblivion needs to be baked to an internal temp of at least 150F. Mine always seem to take longer than specified, not sure if it is due to my bain marie set-up or what.
-The amount of total chocolate solids has to be above a certain level in order for it to set, and milk chocolate has a much lower proportion of chocolate solids (and more sugar).
-Some authors (Shirley Corriher) contend that this style of custard/ mousse cake won’t set if you add too much dairy, due to a pH issue affecting the setting of the eggs. I don’t have any personal experience with that, though, to know if it is the case with Rose’s oblivion. But your milk chocolate would have some dry milk solilds.
All that said, Rose has a milk chocolate variation in the baby cakes section of Rose’s Heavenly Cakes, you might consider looking at that.
FWIW, my kid and most of her friends love the oblivion as written, as long as I use Callebaut dark, the kind that grocers sell in bulk. It’s around 50-53% cacao mass and is mild in style.