First, you need to remember that the US was expecting to use Okinawa as a military, naval and air base for at least two years. None of the commanders involved knew about the Manhattan Project. They were invading Okinawa in April 1945, expecting the invasion of Japan to start in November with an invasion of Kyushu, and the invasion of the largest Japanese island, Honshu, in early 1946. The surrender of Japan in response to Hiroshima and Nagasaki was a surprise to most of the US leadership.
They wanted Okinawa to be a secure base. If there were still active Japanese forces on the island, it could not be properly secure. No defensive line is truly proof against brave men who are prepared to take risks to sneak through it, so there would have always been a few Japanese loose in the base areas. They would also have gone around the ends of the line, by swimming or in small boats. Having enemy soldiers around forces every unit to post sentries, and means nobody can really relax. Having a safe area makes life much easier for troops who aren't on duty.
Further, the troops holding the defensive line would not be available for the invasion of Japan, which was going to need all the available force. It was a very worrying prospect for the US commanders, who knew that they were going to be up against a fanatical population as well as the Japanese military. Estimates of casualties were horrifying, and Okinawa was a reasonable place to put hospitals. Nobody wants enemy troops near those.
The invasion was regarded by both sides as a small-scale version of what an invasion of Japan proper would be like. So it was important for training and discovering the right methods for this kind of combat, and for learning how to minimise casualties.
Finally, the south of the island was where the airfields were, along with the major population centres and the ports. The landing was at the waist of the island, because the beaches there were the most suitable place, but the south was the territory that would serve as bases. Source: Okinawa: the last battle, p. 10. This is the relevant volume of the US Army's official history.
It would be more sensible to ask why conquering the north of the island was necessary, but all of the reasons above about a safe base area apply. This is basic military strategy. It was quite clear that securing all of Okinawa was the right thing to do.