You have asked what is wrong in your analysis. First let me explain the intuition:
You have a sequence X(n). At n=1 the sequence takes the value X(1). At n=2 the sequence takes X(2) and so on. You now define Y(n) as X(-n). This means at n=1, Y(n) takes the value of X(-1), at n=2, Y(n) takes the value X(-2) and so on. Now, you delay the input sequence by k. This means at n=1 the value of the sequence is X(1-k) at n=2 the value is X(2-k) and so on. Now for the same system when this is the input sequence, at n=1, Y(1) = the value of the input sequence at n=-1 which is Y(-1-k). Now can you see the big picture?
There are many ways to understand this: When you take X(n-k) and then flip it as X(-(n-k)) then you are reflecting the input sequence about X=k. But the system reflects the input sequence around X=0.
Now to answer, your question about where your analysis is wrong, let
Z(n) = X( n-k). Then when this is passed to the system the output Y(n) is Z(-n) which is X(-n-k) and not X(-(n-k)) which is X(-n+k).
It is difficult to wrap one's head around this. Atleast it was so for me. But hope it helps.