"so there is a nonzero element $x\in A$ such that $A_x\hookrightarrow B_x$ is finite" I do not follow this part. If $B$ were to be replaced by a field $k$, the claim becomes "given finite type $A\hookrightarrow k$, there is a nonzero $x\in A$ such that $A_x\hookrightarrow k$ is finite". The best I could do is "finitely presented" instead of "finite". See stacks.math.columbia.edu/tag/00FG

@მამუკაჯიბლაძე Note that I wrote "non-zero primes" of $A$, not "all primes" of $A$.