In the Midwest US I also see evangelical Christianity pushing faith healing so hard that to be some kinds of sick are moral failures. Combined with the puritan view of poverty as a symptom of moral failure, and there is a lot of turning up ones nose at the poor and sick as undeserving of public assistance. Only when they bend the knee to God will they be worthy of help, and only with strings attached.
What you describe is more accurately called the "prosperity Gospel", not evangelical Christianity. Mainstream evangelical Christianity doesn't believe anything like what you describe.
What I describe is what I've seen in many evangelical churches over decades and in three different countries, first hand. Evangelicals like to think their flavor is immune from all that ailes other branches of Christianity. Yet they too look down on the sick and the poor, sometimes more than older and more traditional strains of the religion.
Non-prosperity gospel churches don’t say it out loud, but the implications are easy to draw. Massive ideological and cultural structures aren’t logical proofs, there are many contradictions.
But it's the job, the obligation, of a democratic government to create a safety net so people who enter poverty can bounce out of it. No religious nut should be in charge of that, although I can see how that is very convenient to play politics.
I suspect you think that healthcare is helpful, and that if we could only buy poor people more healthcare then their lives would be improved. Healthcare regularly doesn't help that much. Even the richest often die young because healthcare regularly cannot FIX things (think chemotherapy then die).
There's some mental fallacy we buy into that health can be fixed - when the only healthy choice is preventative mitigation. The answer is to live a healthy life: especially regular exercise. But also sensible food choices and removing stress (remove stressors or maybe learn techniques to ignore stress), and avoiding drugs/alcohol/smoking.
Rich Americans are unhealthy too and for the majority how much they spend on medical interventions usually doesn't fix their quality-of-life or length-of-life.
The poor mostly have as much access to a healthy life as the wealthy: so why are both the poor and the rich unhealthy?
The hard question to answer is why we live unhealthy lifestyles ;; is it capitalist advertising ;; should we blame cultural mores ;; are we victims of our own poor choices...
Anecdotes (New Zealand specific):
1: a poorer friend with hip problems that needed replacements in 40s but was forced to live in excruciating conditions for years due to how New Zealand's socialised health budgetary constraints are implemented (via waiting lists). No money was saved by the government because the surgery was eventually paid for (background: surgery cost for both hips was approx equal one pretax year of minimum wages in NZ).
2: two friends with a poor backgrounds that got middle aged diabetes - my guess is caused by booze & maybe smoking. Both THEN started exercise and also changed their habits and lifestyle. One is well off and money didn't really help them. Both could have previously lived healthier lifestyles but only changed their habits after being faced with severe frightening outcomes. Drugs like metformin help (that drug is cheap, however ongoing regular monitoring by doctor is not cheap). Expensive surgical interventions caused by diabetes will never fix the problems caused by poor living.
How much we should blame people's "choices" for poor health outcomes is difficult to answer. If someone chooses to drink booze, how much should society pay for their choice? We have high sin taxes in New Zealand so people have often paid for their medical help however the taxes do not cover the overall societal cost. A better example is acquaintances that have destroyed their health through recreational drug usage (which definitely isn't sanctioned by larger society, and definitely is sanctioned by peer society). I'm a huge believer in personal choice - but I'm no fan of having to pay for the bad choices made by others.
That's the issue with New Zealand's socialism. I can carefully live a prudent life : however it galls to be taxed to pay for people that ignore the cost of their choices.
Aside: it really neeps my bibbles that doctors are seen as saving lives. Engineering, economics and science save most lives. Doctors don't help much. (Ofc "saving" lives is a nonsense term anyway)