Curious as to the reasoning behind why the French favor straining out shallots. Specifically I'm wondering if this is a purely textural thing, or if leaving in the shallots causes the sauce to have too strong of a shallot flavor. Usually I see shallots used when a recipe is looking not to overwhelm the dish with onion-y flavor.
It's a texture thing. Most classic French sauces are strained for texture and appearance reasons. Also, some versions of beurre rouge have more than just shallots in them, making it more necessary to strain lest it be more of the chutney than a sauce.
If you don't care, you can mince the shallots and leave them in the sauce, per Spruce Eats.