A market allows for innovators to find ways to deliver better service for less money and thus offer a more competitive product and gain market share. This competition forces all players to improve, or be forced from the market.
A publicly owned and operated business has no incentive to improve its operational efficiency and little to no incentive to respond to complaints in service.
In the US we have two examples of public insurance providers Medicare and the Veterans Administration (VA).
What we haven't tried as far as I can tell is to apply regulatory pressure with marketing forces. So for example if we made it illegal for pharmaceutical companies from preventing generics to be sold or marketed. If we required by regulation that all facilities publish their costs, and to whom those costs were paid, for every procedure. If we required that all medical procedure costs and options at every facility had to be made available at a publicly searchable database. Etc. It might help in making the market more efficient.
In the UK, the NHS has the incentive to improve its operational efficiency that it simply gets told regularly "we're cutting X billions of your budget. Deal".
The NHS is also split up into a range of trusts that have independent management teams. While there's a lot of cooperation and shared resources, this means there is a lot of opportunity to compare cost and outcomes, and so they can't just hide behind "but that's how much it costs", because the government will know it if there are other trusts delivering similar outcomes at a much lower cost.
And they have the incentive to respond to complaints in service that the NHS Trust in questions board of director will find campaign groups talking to MPs and cabinet ministers, tabloids calling them murderers and the like if they don't keep quality under control, and if they don't respond appropriately, the trust will get put under special measures which can include letting another trust take over management.
There are plenty of ways to apply pressure in addition to markets.
But you have some points: You can certainly make things better by e.g. requiring more transparency.
The trouble with this approach is that we've reached the point where they really can't deal and are consistently blowing through their budgets and talking about refusing non-urgent referrals for months at a time, and the government is just blaming them for overruning their budgets and punishing them with even more budget cuts.