Oort is (mostly) in the solar plane AFAIU, and there's every reason to use gravity assist to get there. I don't know that any specific objects are known, and getting into an Oort-observing orbit might be challenging. Otherwise, yesm it's sparse and arbitrary flybys would likely miss anything interesting.

Spraying random probes in all directions makes lttle sense.

Voyager 1 won't reach the Oort Cloud, if it in fact exists, for another three centuries.