The U.S. did not recognise Rhodesia because it did not want to alienate Britain as well as alienating its own black population. The Johnson administration was weary of getting too involved in Rhodesia. Also Wilson's Labour Government was very hostile to Rhodesia and was comitted to NIBMAR (No Independence Before Majority African Rule), and Rhodesia was not worth ruining the Atlantic Alliance over.
Tthe best chances for U.S. recognition were during the Nixon years. The Nixon administration however was much more friendly to Rhodesia, South Africa and Portugal. In 1971 the Byrd Amendment was passed in the senate that allowed Rhodesian chrome to be imported for strategic purposes (since the Soviet Union was the leading producer at the time). This was against UN imposed sanctions. Air Rhodesia was able to purchase 3 ex-Eastern Airlines Boeing 720 jets in a sanctions busting coup in 1973.
Here are some excerpts from a January 2, 1973 memorandum from the U.S. state department, that illustrate the administration's policy towards southern Africa.
"we made it clear to Africans that we could not afford unlimited economic assistance and that we could neither back the armed liberation of southern Africa nor support measures designed to isolate Portugal or South Africa"
Prior to 1974 Rhodesia was in a strong position. The liberation movements had not made any significant headway and had only made a few raids on the Kariba Dam area. The economy managed to grow and the country attracted 60,000 white immigrants. The Rhodesian Dollar was worth around $1.30 USD and the country was relatively stable. Up until then it seemed that the status quo would live on until the west gave in and just recognised Rhodesia as a fait accompli. However, once Portugal left Mozambique in 1975, Rhodesia was essentially cut off. It's major trade outlets (Beira and Lourenço Marques) were shut off and ZANU had a new front from which to launch an offensive. This in turn led to the west and South Africa pressuring Ian Smith to negotiate a settlement after 1976.