I think the problem with your computation is that, while A is real, its eigenvalues are not necessarily real (we have not assumed that A is symmetric). Hence, in order to test for symmetry, you should be computing complex conjugate transposes (I use daggers for that operation); that is, what you must really show is that

but it is not necessarily true that

You can prove that every normal matrix is diagonalizable, and it is also true that not every normal matrix is symmetric.