I think your premise in the first is wrong. Thrawn gained the high Imperial rank not because of where he originated but because he was so good. Whether only a Chiss could have been Thrawn is a different postulation. If it had been any other race that originally came into contact with Palpatine (or rather, his representative Kinman Doriana, during Outbound Flight, it could have played differently, but Palpatine's humanocentrism was mostly a ploy for the others in the Empire to spend their energy fighting each other and not him. He regarded talent as talent wherever he noticed it.
As I noted above with the Outbound Flight, there were attempts to colonise and explore that region of the Galaxy. It's just that there existed powerful interests in that area who would rather not have had that happen.
But, specifically, in the Old Republic (I don't understand your differentiation of the "Old Republic" and the "Republic" by the way - Ruusan?): too often was the state involved in civil wars, I would say. Especially as there would have been powerful counter-influences who would not be keen on sending people to the 'Unexplored' regions.
During the Empire, again, it would seem that most of the Fleet had to be kept in the known regions taking care of everything which was going on. After all, lots of civil disobedience, etc. Plus, the Moffs would all be unkeen to see another one of their kind gain a bigger power base and would block that move. Palpatine seems to have sent some small expeditions] into the area though -- from the Unknown Regions' Wiki page you posted.. One of these being noted as:
As part of the relations with the First Galactic Empire, the Chiss Imperial Grand Admiral Thrawn, a native of these parts, was dispatched aboard the Imperial-class Star Destroyer Admonitor to the Unknown Regions during the years of the Galactic Civil War, ostensibly on a simple mapping expedition, but in reality to explore and subjugate entire sectors for the Galactic Empire.
Lastly, from a Legends' point of view:
One hypothesis by a noted scientist suggested that a mass of dark matter intersected the galaxy in this region, creating gravitational anomalies that hindered navigation.