No. Your cites don't support your claims at all. Fauci said (then backtracked) that the UK rushed the final regulatory approval of the Pfizer vaccine. Not that the vaccine production and testing had been rushed nor that that the US had rushed anything. And certainly not that there was actually anything wrong with that particular vaccine or any other vaccine that got approval. He was merely expressing surprise that there wasn't a redundant government review of data as there was in the US.
But you didn't prove it. Your source doesn't mention it, mine does. At this point, it's merely conjecture on your part. I cited a source for my claim.