>While we probably should have done a vote:

An election wasn't needed. The Koreans already organized into a popular and sustaining government following Japanese surrender, the PRK. The USMGIK was explicitly tasked with eradicating it in the south, and formally outlawed it in December 1945. Some portions of it remained behind, especially on Jeju Island. These groups ended up protesting the 1948 sham election of Syngman Rhee, who responded with a self declared eradication campaign on the island, resulting in upwards of 100,000 civilians killed, injured or displaced.

In the North, the PRK remained in place and supported by the USSR until the 1948 election of Kim Il Sung, whereby the DPRK was proclaimed and the PRK committees were rolled into the Worker's Party of Korea.

>the USSR which controlled the North would have done a sham vote

There's no evidence of this, quite the contrary. The 1948 assembly elections in the South faced incredibly restrictive voter eligibility requirements. Only landowners were allowed to vote in populous areas, and single elder representatives voted on behalf of entire villages. The election was married by suppression and corruption. There's simply no evidence of corruption in the 1948 northern elections, which were open not just to North Koreans, but South Koreans as well

>A bad border does not justify invasion

Not a bad border, a partition imposed upon a people and maintained by a brutal dictator, his Japanese and sympathizing administrators, and the global imperial hegemon. Following the announcement of separate elections in early 1948, South Korea was already in an open state of civil war, with guerrillas fighting directly against ROK and occupying USMGIK troops. Large scale border clashes had already been commonplace by 1949, with South Korean troops making significant incursions across the border well before the Korean invasion of Korea. In the immediate runup to the invasion, South Korean forces had been stationed at Ogjin, which became one of the opening battles during the invasion. They had also captured Haeju, some 20 miles north of the 38th parallel

The war did not begin in 1950, it began prior even to 1948. The Korean invasion of Korea was simply an escalation in an existing civil war against an occupying force and it's collaborators, not unlike the American Revolution. Hell, the US didn't even care about the South Koreans. It deliberately bombed refugees fleeing from the violence during the 1950 offensive at Nogeun-ri. Some 400 South Korean refugees, nearly all women and children, were bombed and shot deliberately. The incident was covered up until 2001. And a South Korean commission found in 2008 evidence of nearly 200 other similar attacks on refugees by the US military