That would indeed make sense, but as the document appears to have been forged, we must conclude that they did not do this, and we can only speculate as to the reasons why.
My own personal belief is that they did not do this because it doesn't suit their needs, and they had legal alternatives to doing it. I can assume that previous documents are sealed under court order, and while an attorney can ask that the seal be lifted (thereby allowing a print to be made) I believe an attorney can ask that an existing (via adoption) document be modified, nullified, or amended.
That is what I am assuming has happened. They could have asked that all previous adoptions be nullified, but rather than unsealing the original birth certificate, they asked the court to create a replacement using information available off of the original.
Yeah, it's silly, but it would hide the fact that the "Original" was actually a "certificate of home birth" or some such document which will not actually PROVE Obama was born in Hawaii. I have long contended that is exactly what Obama's "Original" birth certificate is; An affidavit of "at home" birth, filled out by his Grandmother with the names of his Mother and alleged father provided by her. (Madelyn Dunham.)
I suspect Barry was actually born in Canada, and grandmother fixed his citizenship problem for him.