My experience of mental healthcare in the UK and China is not what what you imply. Your description of mental care for homeless in USA is close to what is available for the majority of the working class in the UK.

I worked in mental health in the UK. In my area an initial mental health consult has a 6 month wait. You are unlikely to receive ongoing care in the first 2 years after the initial consult, but most will never be offered ongoing care. Lack of treatment pushes people out of the job market and into destructive behaviour. The fact that it is free at the point of care matters little when poor health is itself financially destructive.

I also cared for a relative's health in China. If you want a basic level of hygiene you will pay for everything. If you want clean sheets and avoid bed sores you may need to pay extra. Bring a good blanket, the hospital may not have a heating system.