No, drafts and draft crosses do NOT have to be in shoes just because they are drafts. If their feet are trimmed correctly, they can be barefoot just like any other breed. The most common issue is that their feet are erroneously allowed to get too flared and overgrown to allow them to be functional.
I have personally had several of them, barefoot, no problem. I trim several others that are also perfectly fine as long as their feet are maintained at their genetically programmed size, not at what we "think" they should be.