The others have pointed out the flaw in your suggested argument, but let me discuss something from the first part of your post.

The conclusion of the Baker-Gill-Solovay relativization result (that there are oracles $A$ and $B$ for which $P^A=NP^A$ and $P^B\neq NP^B$) isn't just that "we can't solve the problem of NP=P by relativization," as you say, but rather something more profound: the conclusion is that we cannot solve the NP=P problem *by any method that admits of relativization*. This is an enormous class of methods, which includes all of the standard powerful methods of computability theory. The reason is that if we could prove $P=NP$ or $P\neq NP$ using methods that can accommodate oracles, then we would immediately deduce the corresponding equality or inequality for all oracles, in contradiction to the Baker-Gill-Solovay result. 

The significance of this is therefore that since all the standard methods do admit relativaization, we will not be able to settle P versus NP using only those methods; we must be more imaginative and subtle. Any purported proof answering P versus NP must make use of proof methods that do not relativize to oracles.

The point isn't that the theorem rules out relativization as a method of solving P versus NP, but rather that it rules out any method that admits of relativization. Since this includes most of our methods, it is a serious obstacle.