Currently most countries that use a form of Universal healthcare are rapidly losing money every year.

Healthcare in most countries with public healthcare systems costs less than in the US both in terms of public and private spending. Ironically, despite our "socialist" economy, most countries in Europe have a better budgetary and fiscal discipline than the US.

Germany, for example, is running a budget surplus while Netherlands has a balanced budget.

(2) the government can be trusted to allocate the healthcare fairly

Can private corporations be trusted to allot healthcare fairly? In terms geography, not so much. Where there's no profit, there's no incentive to open a practice and that's absolutely logical.

Government healthcare systems compensate for lack of profitability in peripheral regions through "allotting" healthcare to the regions where it is lacking, giving doctors incentives to move there, creating better working conditions, tax breaks etc.

This is a big issue in rural places in Europe these days.

Other than that, the universal healthcare works in a similar way to private practices. In most countries, the centralized national insurance company makes contracts with private doctors and hospitals.

In addition, there's nothing preventing someone from running a fully private medical service or a practice.