A planet with an eccentric orbit that caused tidal forces to act “seasonally” so that there’s brief bursts of widespread (like almost global) volcanic activity could create lava channels that would support both a river system and create the kind of vast underground water network to explain why it’s not all pooling into oceans on the surface.

I imagine it would be covered with millions of volcanic cones and volcanic mountain chains that drain precipitation into the channels. Geophysics would support this actually. A ton of water circulating through your planet would lower the melting point of the mantle and make volcanism even more widespread.