I think, the reason why one cannot define these Hecke operators is that they somehow 'switch' the characters, i.e. if one decomposes
$$M_k(\Gamma(N)) = \bigoplus_{\chi} M_k(\Gamma_1(N), \chi)$$
where now, $\chi$ runs through the additive characters, one could expect that for a modular form $f \in M_k(\Gamma_1(N), \psi)$, interpreting this modular form as a modular form for $\Gamma(N)$ and then applying the Hecke operator for $\Gamma(N)$ should be the same as applying the $\Gamma_1(N)$ Hecke operator on this $f$. For this reason, i believe that one should define the Hecke operator on the whole space $\oplus_{\chi} M_k(\Gamma_1(N), \chi)$ as the pullback of the $\Gamma(N)$-Hecke operator. This means that the $f$ above is interpreted as a vector $(0, ..., 0, f, 0, ..., 0)$ and the Hecke operator 'mixes up', i.e. it is possible that if $g = T(m)f$ as $\Gamma(N)$-modular form, the result $g$ has a decomposition that corresponds to some vector $(g_{\chi})_{\chi}$ where some of the $g_\chi$ for $\chi \neq \psi$ are nonzero.