White Lily Flour in a Roux
I live in Canada and don't have access to WL. When I'm at my inlaws for Thanksgiving I get the task of making gravy. Every year I seem to have trouble with getting the desired thickness when I never do at home. The only real variable is the flour. I use WL down here.
Does WL not thicken as well as other, harder, AP flours??
It's the starch that provides the thickening, not the gluten forming proteins. WL is 'softer', lower gluten, so it (probably) has more starch than the harder Canadian AP. Lets say 8% protein for WL, 12% for Canadian. While that is a big change on the protein side, the starch difference as not as significant. 88 v 92 % (ignoring other components like moisture).
The numbers suggest that the Canadian AP doesn't thicken quite as well as WL, but not to the extent that most cooks would notice.
"What's the white 'floury' substance in this Zip lock bag maaaaam?"
"Oh it's just 'White Lily' flour for making gravy officer".
"Is that what they call it where you come from? The name sounds sort of, well, 'foreign' right?"
"Ya but, ya but".
"Turn around maaam. You are not under arrest. We are handcuffing for our protection until we can have your 'White Lily' tested. The officer who can do that won't be in until the day after tomorrow. Just follow the officer." LOLOLOL
"Cake flour has about 20% more thickening power than bread or AP (All Purpose) flour."
Which conflicts with your experience (WL being more like cake).
Are you using the same type of fat in both places? Butter is about 20% water.
Different size and shaped pans might affect your judgement, especially if you work more by appearance and feel than exact measurements.