Sensor beams seem to act like a faster than light radar, but I'm sure you'll agree you can't look at distant solar systems that way, any more than you could light up the Superdome with a flashlight. (And an astronomer's laser is lucky to bounce one or two photons back from the moon.)
When going into uncharted space, I propose rigging an unmanned, warp-drive shuttlecraft as a recon drone. Let it travel quickly to surrounding star systems while the Enterprise sits in the middle of no place.
When the drone comes back, Kirk reviews its data and decides where to fly the Enterprise to. He'll save tons of warp drive fuel that way, because he won't have to push the mass of the huge starship to every solar system in sight. The little sensor drone will burn a fraction of the fuel.
And the purpose of Franz Joseph's telescope-like device remains a mystery.