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

$\displaystyle A^{\dagger}=(P^{\dagger}DP)^{\dagger}=P^{\dagger}D ^{\dagger}P=P^{\dagger}DP=A,$ but it is not necessarily true that

$\displaystyle D^{\dagger}=D.$

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