Britain was never in the history of the last centuries really independent. It either was part of a large empire or part of EU and NATO.
Currently all governments and the majority of EU parliament have to agree to decisions. A simple French-German agreement is not sufficient. For sure, the EU should be more democratic - but UK was one of the states which blocked that in the past, because most British governments wanted only an economic, but not a political union. The UK had not even joined Schengen (which was extremely annoying when traveling to UK). Now UK has the choice between paying to get access to the market without any possibility to participate in decisions - and not have easy access to the markets. There is no saving money from leaving the EU, because also the EU support for UK (e.g. for research, farmers, communal infrastructure) will be removed in addition to the economic costs of losing privileged access to the markets.