Does a unitary transformation change the symmetry of a system?

Assuming that I got a system with hamiltonian given by$$H=sigma_x+sigma_y$$where $sigma_{x,y,z}$ denotes the Pauli operator. It’s not hard to see that this hamiltonian has chiral symmetry respect to $sigma_z$,i.e.,$$sigma_zHsigma^{dagger}_z=-H$$Now I want to make a time-dependent transformation $U$ to the state of system,i.e., $$left|Psi right>=Uleft|psi right>=e^{-isigma_0t}left|psi right>$$where $sigma_0$ denotes the $2times 2$ identity operator. Under this transformation, hamiltonian H must be tranformed into $widetilde{H}$,$$widetilde{H}=UHU^{dagger}+ifrac{partial U}{partial t}U^{dagger}$$ in order to satisfy with the Schrodinger equation$$ifrac{partial left|Psi right>}{partial t}=widetilde{H}left|Psi right>$$ It’s easy to obtain $widetilde{H}$ with form given by$$widetilde{H}=sigma_0+sigma_x+sigma_y$$and as you can see, the chiral symmetry was broken because$$sigma_zwidetilde{H}sigma^{dagger}_zne-widetilde{H}$$due to the factor $sigma_0=ifrac{partial U}{partial t}U^{dagger}$. What’s wrong with my calculation or my understanding?