This is not an argument for or against anything here, but it must be pointed out that correlation does not equal causation.
Germany, Italy, France, Israel, Iceland, South Korea, Switzerland, Finland, Hong Kong and Singapore are all examples of successful, wealthy and high QOL societies that do not have a monarchy. IMO their success and progress has little to do with their monarchical status and more to do with their culture, people, history and other complex socioeconomic factors. By that logic, I would also posit that the success of the monarchies above has little to do with their ruling family and much more to do with...again, culture and other economic and sociological factors which were not influenced by their crowned heads
And I'm pretty sure the NHS did not originate as a result of the BRF nor does its continuation depend on there being a crowned head of state in the UK. Also, the cost of a President in the UK cannot be compared to the US or France where the President is the head of state and the head of government. In countries where the President is simply a figurehead, the costs are much lower and there is also the issue of having to pay security for the President's direct nuclear family and not an extended lot like the BRF (although that has improved in recent years).