outhern Manchuria and the Korean peninsula were fragmented among numerous culturally and linguistically related tribes, and your suggestion is similar to saying that it wouldn't have been a problem for the Roman Empire to occupy Germania. In both cases, although it might have been possible to occupy a significant amount of territory, it would have been difficult, if not impossible, to take over the entire area because of frequent raids and occasional uprisings.
At best, Han control was limited to the Liaodong peninsula and a small portion of northwestern Korea . . . two of the four commanderies were lost within 25 years, while another's (Xuantu) capital was lost within that time period, and its borders were shifted further north. This suggests that another entity in the area, most likely Goguryeo, had raided the commanderies to the point where the Han thought that it was unprofitable to retain all of them, and were content with a looser control over the region as a whole. The Samguk Sagi also suggests that the Lelang commandery, which lasted until 313, was briefly conquered around 30 AD before the Han recaptured it . . . there was a prolonged power struggle between the Han and the natives for over 300 years, and this would need to be completely butterflied away in order for the Han to even think about expanding into the southern part of the peninsula.
[T]he Han was already preoccupied with the Xiongnu for over 200 years, and in fact, the Han invaded Gojoseon because it had allied with the latter, and had also cut off trade relations. The prolonged conflict, along with other ones to the west and south, would be much more important to the Han than taking over fragmented states in the peninsula, then invading what is now Japan in order to conquer more fragmented states. Also, by the time that the Xiongnu collapsed as a whole in 89 AD, Goguryeo would have been developed enough for the Han to just retain loose control over them, as it learned in 172 during the Battle of Jwawon.