Wouldn't it be easier to cook the beets before peeling and just slip the skins off, then dice?

Like Amanda said, the orzo gets cooked in the beet water. This gives the pasta a vibrant reddish orange color. If you boil the beets with their skins on, the water may not become as tinted, and flecks of skin might boil off the beets...which would end up in your pasta.
But, you can skip this part too, and use a separate pot for the orzo. The pasta won't be red, but I'm sure it would still be tasty.

Sure, you could do that but you want to make sure the beets are well scrubbed before boiling because you cook the orzo in the same water after the beets are taken out.