There's not much, in every direction, for a long ways.

It makes more sense, generally, to send probes to where things (planets, moons, dwarf planets, asteroids, comets) are, that we can reach in reasonable time (years, possibly a decade or more for outer targets).

"Every direction" just takes you to deep space. And that's not going to be particularly illuminating, though a few such probes (say, a high- or low-solar plane one) might be interesting.

Oort cloud is still not confirmed, right? I get that it might be incredibly sparse, but isn't there anything to study there?

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.

I think you're mixing up the Kuiper Belt with the Oort Cloud. Kuiper Belt is in the plane, and within the Oort cloud. Oort cloud is a sphere.

At least, according to ESA [1].

[1] https://esahubble.org/images/opo0204i/

Hence my hedge, mostly.

Though this is well outside my area of expertise.

I was thinking it would be interesting to send a stream of them out in one direction (or many streams) with the idea that they could serve as a relay/mesh network, allowing us to communicate with the probes (and vice versa) for longer. Not sure how valuable that would be scientifically.