You're right that the budget is tighter, however the healthcare budget is separate from the main state budget, and you can't have a situation where for example you'd allocate less to health to pay for schools or army or something like that. The money still comes from taxes though and there is enormous pressure currently due to the economic situation in Europe, especially from businesses, to lower those taxes. It can be sometimes hard to get people to defend this system when they have never had a big health concern before.

However when you switch to a for-profit healthcare then the interests of the operators stop being aligned with those of the population. Even if there is more money, like in the US where the budget per person is about double that of France, it is not necessarily spent in a way that benefits the patients. State-run health avoids multiple administrative costs due to multiple actors, it avoids having to pay dividends to share holders, and as a single payer it has much more power over the pharmaceutical industry to lower costs. Overall it is a more efficient use of people's money.