Sauce – Why use clarified butter in a bearnaise

Your recipe is suspect. Melted butter is the same in chemical composition as softened butter. The two react a bit differently in recipes due to the way that they interact with the other ingredients to impart textures: in short, the creamed butter will hold microscopic bubbles that the fully melted butter will not. Adding egg yolks will help the melted butter incorporate, and probably lead to more cakey cookies (compensating for the flatness you get from melted butter), but the effect is hard to isolate.

As for brown butter, what happens is that as moisture evaporates from the butter, the boiling point increases (similar to what happens when making caramel or candy). As the boiling point of the butterfat increases, the temperature of the butter solids (which consist of lactose and milk proteins) begin to caramelize (the sugars) and undergo a maillard reaction (the protein). Both of these reactions will impart unique flavors to the butter, yielding the distinctive brown butter taste.

As the butter heats, the integrity of the fat is damaged. This occurs to all fats, which is why you can only use a batch of fat for deep frying a few times before it needs to be discarded.

You can absolutely do this. I do it in brown-butter cornbread, for example. A good trick to boost the flavor even farther is to add non-fat milk powder to the butter as it browns, to supply even more protein for the Maillard reactions.