I don't think the UK has ever had a government that actually wanted the NHS to fail. Even the most right wing conservatives realize they're stuck with it and that voters will be happier if it works better. They have very few levers to achieve that. Private sector involvement is one but not a very powerful one given that the NHS killed off the private health sector in the UK and ensures it can't regrow to any kind of meaningful capacity. To the extent it grew in the last few years that's only because the NHS is now in the stages of advanced collapse, failing to provide even the most basic services you'd expect and therefore for some paying twice is the only way to get healthcare at all.

I also really doubt there are many in government or politics who want a US style system. The US has a strange system replicated nowhere else (much like the UK). The way healthcare is tied to the employer there is a legacy of socialist economics during WW2. To try and stop inflation wage controls were implemented so companies started to compete via non-wage benefits, like bundled health insurance. Then for reasons I don't know it was made tax-exempt, so the tax system incentivized workers to demand healthcare through their employer instead of paying it themselves. And then legal changes and union campaigns cemented that system. Part of why US healthcare is so expensive is the sheer number of layers between the people who use it and the act of paying for it. It's very much an accident of history born in the war, and not a model to obviously replicate.

The UK system is likewise rather dysfunctional thanks to WW2. Years of wartime propaganda followed by a victory convinced Brits that the government must be very good at things, and of course this was an era in which socialist economics was taken seriously across society. So on victory the UK immediately voted in a Labor government that nationalized many industries including the entire healthcare system, with Bevan famously dividing and buying off doctors who realized it'd be a bad idea by "stuffing their mouths with gold". Governments have attempted to use the same strategy to solve its problems ever since, yet have never been able to predict or control costs in the way a business would need to. Even in its very first year, costs were double what was predicted. This is very far from the original belief that GDP improvements from the NHS would be so great it would effectively pay for itself.