I dont understand what the reasoning could possibly be. Such a small number of people could have been given UK citizenship plus a massive sum of cash, and the UK still would be in profit.
The reason why the British left Qatar in 1971 is not related to oil, but rather to the decline of the British Empire in the post World-War II era. I'll explain my answer down below by first explaining how oil was found in Qatar and how the British left the country in 1971.
The British were involved in the Persian Gulf since the early 19th century when it fought local pirates on trade routes and established peace treaties between the sheiks of the region who were battling each other at sea. In exchange for the stability of peace and protection, the British, like it did in many other countries, signed a treaty with Qatar that gave the control of the foreign policy of the Gulf State to Britain.
The Anglo-Iranian Oil Company (BP today) in which the British government held a 51% share, obtained a concession in Qatar in 1925. In 1928 the Red Line Agreement was signed, to make a long story short, this agreement basically divided the oil fields of the former Ottoman Empire between the victors of World War I (The British: Anglo-Iranian, the French, the Americans, and in some respect the Dutch through Royal Dutch Shell). The British had the most "influence" when it came to directing how the oil in the agreement should be managed (It's a lot more sophisticated than that but I'll be happy to explain the Red Line Agreement in a different question). A new company was created called the Turkish Petroleum Company, but that name was changed to the Iraq Petroleum Company (IPC) after oil was found in Iraq. IPC was the subsidiary company of all the participants in the Red Line Agreement. The Qatar concession was managed by subsidiary company of IPC: Petroleum Development (Qatar).
Sidenote: During and immediately after World War I, the control over foreign reserves was achieved through two different steps. The first one was via jointly owned subsidiary companies created by larger corporations who joined forces and assets to exploit oil in a certain location; the seven biggest companies in the Middle-East at that time were all connected to each other by jointly owned subsidiary companies. Once a subsidiary was put in place, it was time to negotiate for a long term contract; because of the amount of oil involved, the volatile nature of oil prices, the sale conditions, and provisions that restricted the marketing of certain quantities of oil, long term contracts provided more assurance and were therefore essential for big oil companies.
In 1940 Petroleum Development (Qatar) discovered the Dukhan oil field, the largest one in Qatar with an Estimated Ultimate Recovery (EUR, An estimate of the total amount of oil or gas that will be recovered from a field over its entire lifetime) of about 1.6 billion barrels and 9 trillion cubic feet of natural gas. World War II put a halt to operations. Production did not begin until 1949, at a level of about 2,000 barrels per day. It grew rapidly, surpassing 100.000 barrels per day by 1955.
In the 1960s, the British were in a state of decline on somewhat all fronts, economically, politically, at home and abroad, it became clear the British Empire of old didn't have its place in a post WWII era. In general, words and concepts like post-colonialism, imperialism and economic exploitation were denounced worldwide. The Vietnam war for example is a perfect illustration of this new wave that provoked a crisis for the United States in South-East Asia, but also at home during the College protests of the 1960s. In early 1968 in response to a payment crisis, Prime Minister Wilson announced that Britain would end its defence obligations East of the Suez Canal by 1971 eliminating in the process British presence in the region. Qatar subsequently declared its independence on September 1st 1971.
Between 1974 and 1977, in reaction to the Yom-Kippur war and like many other countries in the region, the government of Qatar purchased the Petroleum Development (Qatar) in two phases. It created its national oil company, the Qatar General Petroleum Company (QGPC), in 1974. QGPC acquired the Petroleum Development (Qatar) in 1977, completing the nationalization of the sector.
Though oil and politics are closely intertwined in the region. The reason why the British left is not really related to oil but rather to the inevitable decline of the British Empire and the necessary costs both financially and politically to maintain it.
My field of study is more related to the early history of oil in the Persian Gulf region (Saudi-Arabia to be precise).
if you have any additional questions, feel free to ask them.
Bibliography
Vassiliou, Marius S. Historical Dictionary of the Petroleum Industry. Scarecrow Press, 2009.
Yergin, Daniel. The Prize: The Epic Quest for Oil, Money & Power. Simon and Schuster, 2011.
Smith C., Simon Britain’s Revival and Fall in the Gulf: Kuwait, Bahrein, Qatar and the Trucial States, 1950-1971. Routledge, 2004.