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You could just glue the mirror onto the board, as I do at the end, without the recess but I'd worry about the possibility of 'glue creep': the weight of the mirror quietly pulling down against glue that's softening in warm weather. The mirror here is lightweight and the glue an epoxy resin, so very unlikely, but I'd always be checking! Recessing the mirror gives its weight something to sit on.
Another reason for recessing is aesthetic: I like the edge of the mirror to be hidden with just the bevel showing to catch and reflect light. You could turn the circular recess or route it out, but since many of you don't have these machines we'll do it the easy way. You'd use a similar approach if the mirror was square or oval, say.
Don't make the recess too tight; you should be able to spin the mirror easily. Notice that the carving here - lowering, levelling and setting in the edge - is exactly the same method you'd use for any low relief woodcarving, so a good little exercise in it's own right.
