My question is about why does the result of Baker-Gill-Solovay not prove that $P \neq NP$. There have been several questions on this forum about this topic perhaps but I couldn't find my specific question amongst them.
Context: This is what I had in mind when I first saw the Baker-Gill-Solovay result (there exists an oracle $A$ such that $P^A \neq NP^A$): I will try to give a proof for $P \neq NP$. Suppose $P=NP$ then $P^A = NP^A$ for all oracles $A$. By Baker-Gill-Solovay result we arrive at a contradiction. When I asked my prof about this he said that the implication $P=NP \implies P^A = NP^A$ is wrong. So I wanted to check why (and where) does it fail.
I think it is correct that $P \subseteq NP \implies P^A \subseteq NP^A$ for all oracles $A$ because any poly-time TM with oracle access to $A$, is also a poly-time NDTM with oracle access to $A$.
My question is: Why (if $NP \subseteq P$ then for any oracle $A$, $NP^A \subseteq P^A$) is false?
My reasoning would go as follows: $NP \subseteq P$ means that given any NDTM (which involves a guessing sequence, followed by a poly-time verification), I can replace the guessing sequence with some poly-time TM. But given a NDTM with oracle access to $A$, I cannot directly replace the guessing sequence with some poly-time TM (as per the hypothesis), because in this NDTM, the guessing sequence might involve the access to oracle $A$.
Following that, I want to argue why isn't it so, that given a NDTM with oracle access to $A$, I can move all these calls to oracle $A$, after the guessing sequence? That is to say - why couldn't I, using non-determinism, guess the answers to my oracle queries and in the final step, verify these particular guesses using actual oracle queries?
More precisely, can I not re-define the definition of $L \in NP^A$ as follows: there exists a poly-time TM $M$ and a polynomial $p$ such that $x \in L \iff \exists w$ such that $|w| \leq p(|x|)$ and $M^A(x,w) = 1$?
Apologies for being too vague in my arguments above but I have just started learning Complexity Theory.
Edit1: Changed the question from given a oracle $A$, if $NP \subseteq P$ then $NP^A \subseteq P^A$ to if $NP \subseteq P$ then for any oracle $A$, $NP^A \subseteq P^A$
Edit2: My definitions are: $P^A$ is the set of languages decided by a poly-time TM with oracle access to $A$ and $NP^A$ is the set of languages decided by a poly-time NDTM with oracle access to $A$
Edit3: Added more context to the problem.