The Battle of Okinawa was one of the bloodiest and most difficult of WW2. But, if I understand correctly, most of the casualties were incurred during the fighting for the southern part of the island. Considering that the sole use of Okinawa for the Americans was as a base and stepping stone for the invasion of mainland Japan, why was the capture of the south necessary?
At the time of the battle, the Japanese garrison had no hope of resupply so the Americans could have just captured Yontan airfield and enough territory around it to prevent artillery attacks and then let the Japanese either starve in their positions or waste themselves in hopeless attacks against a defensive line. Why did the Americans insist on an extremely difficult and, to my mind, pointless capture of the southern part of Okinawa?
There are some misconceptions in the question which need to be cleared up, and doing so will go some way towards answering the question as posed.
The following is sourced from the official US military history of the operation which can be found online here:-
Firstly, the specific purpose of the mission as conceived and planned was primarily to secure the southern part of the island, as this was where the best roads, the major port on the island, the best anchorage, the existing airfields, and the terrain deemed most suitable for further airfield construction were all located. It was also where three quarters of the population were located, and it was hoped that some of these might be employed as a labour force.
Secondly, the planned US Okinawa military base was not intended solely for supporting the invasion of Japan, there were also potential operations against Formosa and the east coast of China under consideration which were also to be supported from Okinawa, should they eventuate. One secure airbase was far less than what the Americans were hoping to achieve here.
Thirdly, and probably most significantly in answering the question, US intelligence relating to the terrain of Okinawa was very limited prior to the operation, and it was expected that only 8 airfields would be developed on the island. After examining the island at first hand it was found that 18 airfields could be sustained, including those suitable for long range bombers, so the rugged northern part of the island had been neglected as strategically useless prior to close examination, and planning of the operation reflected this. The perceived lack of strategic value of the north was also shared by the Japanese as evidenced by their decision to barely defend the northern part of the island themselves.
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.