Integral operators induced by symbols with non-negative Maclaurin coefficients mapping into $H^\infty$

For analytic functions $g$ on the unit disc with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator $T_g(f)(z)=\int_0^zf(\zeta)g'(\zeta)\,d\zeta$ from a space $X$ of analytic functions in the unit disc to $H^\infty$, in terms of neat and useful conditions on the Maclaurin coefficients of $g$. The choices of $X$ that will be considered contain the Hardy and the Hardy-Littlewood spaces, the Dirichlet-type spaces $D^p_{p-1}$, as well as the classical Bloch and BMOA spaces.