The strategy you describe - isolating and bypassing Japanese island garrisons until they become incapacitated from lack of supplies - is in fact the very strategy that the U.S. did pursue in the Pacific war. Called the "Island Hopping" strategy, it involved bypassing fortified Japanese-held islands, and using allied air- and sea-superiority to prevent Japanese re-supply or re-deployment. Some of the notable island fortresses bypassed and isolated in this way include Truk and Raboul (home to very large Japanese garrisons), as well as Chichijima (where future president George H.W. Bush was shot down and rescued by submarine).

As the war progressed, the U.S. was able to flood the Pacific theater with ships, submarines, and aircraft in such numbers that resupplying island garrisons became nearly impossible for Japan. By that point, the U.S. could essentially pick and choose which battles to fight and when.

But even with an island hopping, you can't hop over every island - you have to fight the enemy somewhere. For example, some islands had to be seized in order to construct the forward airfields and harbors required to keep the operation moving ahead. Other island battles were fought because the U.S. wanted to deny airfields/harbors to the Japanese, or as an emergency landing location for damaged U.S. bombers returning from missions over Japan itself (i.e., Iwo Jima). In hindsight (as well as at the time), the decision to attack rather than bypass Japanese islands proved controversial. For instance, some historians believe that the land fighting on Iwo Jima, Bouganville, Peleliu, and the Philippines did little to advance U.S. strategic aims and could have been avoided.

Getting back to your question, Thin Red Line takes place on the island of Guadalcanal. The Guadalcanal campaign occurred early in the Pacific war, when the Japanese were still on the offensive throughout Asia and the Pacific, and seemed nearly unstoppable. The Guadalcanal campaign represented the first U.S. counter-offensive of the war. It was largely motivated by the discovery that the Japanese were constructing an airfield on the island, in support of their own planned further offensives. At this stage of the war, the Imperial Japanese Navy was a deadly and formidable adversary, operating in strength around the island. For this reason, U.S. planners felt that the island had to be attacked directly, and sent the Marines (and later Army, as portrayed in Thin Red Line) to seize it.

The Marines captured the airfield early in the fighting, and the Japanese were never able to take it back, despite ferocious and desperate efforts. The resulting allied air-superiority made the campaign essentially unwinnable for the Japanese (most Japanese soldiers on the island died of disease and starvation because re-supply had become nearly impossible), and they eventually evacuated their remaining troops and abandoned the island to the Americans. From that point forward, U.S. power was more and more ascendant, and the island hopping strategy proper could begin.