Creative telescoping and generating functions of (variants of) multiple zeta values

We show how to convert the generating series of interpolated multiple zeta values, or multiple $t$ values, with repeating blocks of length 1 into hypergeometric series. Then we invoke creative telescoping on their generating functions, in some known cases for illustration, and in some apparently new cases, reducing them to polynomials in Riemann zeta values. The new evaluations, including $\zeta^{1/2}(\{\bar2\}^n,3)$, $\zeta^\star(\{1,3\}^n,1,2)$ and $t^{1/2}(2,\{1\}^n,2)$, resolve some questions raised elsewhere, and seem to be non-trivial using other methods.