The idea of wiki edit suggestion privileges is to substantially improve it. Therefore, my suggestion would be that it might be okay to use the Tag Wiki Excerpt inside the full Wiki, but only if you plan to substantially improve it.
Substantially improving a tag wiki involves adding example questions, listing situations when one might plan to use the tag, when not to use the tag, and showing examples of the topic that the tag discusses.
If I saw a Tag Wiki that was simply copied from the excerpt, I would reject it, unless more details are added the qualify as a substantial improvement. However, I can't remember at the moment if the Wiki and the Excerpt are shown together.
They are in fact separately approved.
Additionally, I believe that adding the excerpt to the Tag Wiki itself, either manually or automatically, could marginally discourage other users from improving it further. I recently made improvements on a tag excerpt that had no wiki whatsoever and left the body incomplete with a few links. I personally felt like leaving the main Tag Wiki incomplete motivated me to come back later with more research, and fill in all of the missing details.
A gaping, empty hole screams "improve me" a lot louder than something that already has some content in it, IMHO.