Republic of Minerva looks good. It fails in one thing. It fails to protect itself.
Capitalism is good for pretty much everything but security. So a rich capitalist country paying US, UK, or China for protection seems like a good idea.
There are a number of Protectorate States with good relations with their protector state. For example, Butan (India), Lichtenstein (Switzerland), San Marino (Italy and France(?)), Monaco (France), Niue (New Zealand), The Cook Islands(Also New Zealand), and Palua (United States). The reasons for many of these are unique and varied throughout time and origin and really are decided by historical and political reasons rather than economical ones (San Marino, Monoco, and Lichtenstein are rather wealthy nations).
More common is what is known as Protecting Powers and there are some rather big countries that are the beneficiaries of Protecting Powers. Protecting Powers are basically when one state is a protectorate of another but only with respect to certain foreign relations. Almost always when two countries cut of diplomatic ties with each other, they will name a third nation with extant relations as a protecting power. Switzerland is probably the biggest country that uses protecting power authority as their neutrality means that they tend to get picked to play telephone. During both world wars, Switzerland basically was handling the Allied Nations negotiations with the Axis Powers on the other side of the war. Typically Protecting Powers means that the nation being protected will have a small office in the Protecting nation's embassy to the hostile country... though in some cases, the protecting nation does diplomacy on behalf of the protected. For example, until recently, the United States had a small office in the Swiss Embassy to Cuba (US and Cuba having no formal relations, Switzerland would be the Protecting Power of the United States to Cuba) while the US definitely does not send personnel to the Swiss Embassy in Iran. Cuba (I think) used Switzerland as their Protecting power to the US and had offices in the Swiss Embassy in the United States. Iran uses Pakistan as their protecting powers and I'm not sure about if they have an office section for Iranian Personnel in the Pakistan Embassy in D.C.
Protectorates are dependent territories - they don't have full political independence. People generally prefer to have a say in how they are governed so other than micro states, such as Monaco and the like, most protectorates were either absorbed or gained independence.
There is not much incentive on the protector side either. How do you tell your citizens that you are sending their sons and daughters to fight someplace because someone paid you $X billion? Not only it denigrates your country to a mercenary state, it's also immoral and politically infeasible.