I wouldn't use beurre manie in any sort of demi-glace, whether made by meandering down the espagnole path or as a more or less straight reduction. If a demi isn't sufficiently structured or thick enough, you just keep reducing. Maybe on some sort of demi derivative, if you didn't want to over-concentrate the flavors.
Beurre manie is just one way of adding structure (as well as thickness) to a soup or sauce. It's the right way for some things, and it's a convenient bandaid for a lot of others. It doesn't bring much flavor -- or at least it shouldn't.
It stands up to heat better than arrowroot, to time better than cornstarch, potato startch, is more predictable than tapioca startch, and so on. For that matter you use Wondra for most of the same things. It's alright. Real chefs do.
Beurre manie (and Wondra too) have their limits as bandaid thickeners. Yes, they thicken and yes -- if you use a lot -- they can do a lot of thickening. But they aren't "foundational" in the same way roux, tomato paste, reduction, or dissolved onions are. The taste contribution of raw flour added near the end of cooking will almost never be helpful.
Escoffier is great stuff, but needs some context. For instance, in the beurre manie matelote example -- it might help to know that a matelote gets its taste and color from ground lobster shells. You can't thicken too early or the shell won't cook down, you can't use cornstarch because the wine would break it, arrowroot couldn't handle all the heat, and so on.
My example -- and I'm just crushed y'all aren't talking about -- go for similar results for different reasons. I chose beurre manie to allow the flavors to marry and develop before the sauce was thickened, and for the control beurre manie brings. I didn't want something as thin as a Carolina sauce, nor as gloppy as KC style, just a nice jus made by someone using technique in service to control.
In my opinion it's a big mistake to use it to replace all other thickeners -- but I'm not in the OP's kitchen. In my own cooking, I would never use it to replace roux. Nor would I ever make an ersatz with oil. Furthermore, It's a big mistake to try to impose theory on reality, and better to work the other way around. Since cooking is results oriented, if it's working for him -- and I take his word for it -- whether or not it works for me, I'm thrilled.