"The US could have dropped the bomb off the coast of Japan as a
demonstartion"

Timeline for Bobert:

On July 26, 1945, Truman and other allied leaders issued The Potsdam Declaration outlining terms of surrender for Japan. It was presented as an ultimatum and stated that without a surrender, the Allies would attack Japan, resulting in "the inevitable and complete destruction of the Japanese armed forces and just as inevitably the utter devastation of the Japanese homeland".

Prime Minister Kantaro Suzuki declared at a press conference that the Potsdam Declaration was no more than a rehash of the Cairo Declaration and that the government intended to ignore it. The statement was taken by both Japanese and foreign papers as a clear rejection of the declaration.

No surrender.

On 4 August 1945, American aircraft dropped leaflets on Hiroshima warning the citizens to expect terrible destruction to be visited upon their city because Japan had refused to surrender. Although many civilians had already been evacuated to the country, this warning was largely ignored. On August 6, the first atomic bomb was dropped on this city. At Hiroshima, 60,000 Japanese died and a similar number were injured.

Still no surrender.

Three days later, when the first atomic bomb had still evoked no response from Japan, a second bomb was dropped on Nagasaki, a port with naval installations. The primary target on this day had been the city of Kokura where a huge army arsenal was located. Thick clouds over Kokura forced diversion of the B-29 with the second bomb to Nagasaki. At Nagasaki, 36,000 were killed and about 60,000 wounded.

Japan Surrendered.

PS:
80,000 people died in one conventional bomb attack on Tokyo on the night of 8/9 March 1945 than in Hiroshima the bombing.