in what sense is to interpret the sentence the schrodinger operator has infinite propagation speed. And what about the fact that if there is a singularity at $t=0$, then at $t=\epsilon$ there isn't anymore?
Consider the Schrodinger equation with an initial condition supported in a bounded region. At any positive $t$, the wave function is nonzero arbitrarily far away. In that sense, the particle can travel (albeit with small probability) at arbitrarily high speeds.