The fundamental problem is that the narrative of the current administration is that globalization and global co-operation has been bad for the United States and that this policy has resulted in the U.S. being taken advantage of.
However, the correct response to this pandemic would go against this narrative completely. It is a common foe faced by all nations, and is an ideal situation for being tackled through global co-operation!
The current administration is continuing to try to solve this problem without turning to global co-operation. If the vaccine is developed in Germany, or the UK or China, the U.S. would then need to license this technology, and it would reveal that the U.S does in fact need the world, and that global co-operation is, at the very least, needed in some situations. It would lose face. If on the other hand, the administration can claim "Look! We developed the vaccine here in the U.S. .. I told you we didn't need those other guys. We will take care of Americans first, and then we will help the others." then it would validate the "America first" narrative.