You conflate making people pay for health insurance with them having access to healthcare. That's real sophistry.
It's not sophistry, because it's true.
The typical way this works is that people on low incomes tend to be better at hitting forms of core, planned payments: rent, bills, major insurance, etc. Then they meet ongoing expenses of necessities, and whatever other luxuries they can afford that fills the rest of their income. Without cover, healthcare ends up in the latter category. Someone feels unwell, they decide whether it's worth affording the medical fees: after all, it may just be a passing illness and if they wait it may go away. Paying an unexpected bill they haven't planned for becomes highly undesirable, especially for what might be a transient sickness. (It might also not be transient, and those weeks-months they put it off could result in severe harm or death.) However, as insurance, healthcare costs move into the former as a planned expense. Once they have cover and the risk of being hit with substantial out of pocket expenses diminishes, instead when they feel ill they seek care. Thus cover facilitates access.
To add to that, where healthcare costs are particularly high, without insurance people often just don't get treated, or get substandard treatment, because the medical provider won't get paid. Sure, they'll patch up an emergency case, but what then? Without payment, nothing. Have you ever watched US medical shows? Did you ever wonder why in shows like Scrubs they featured storylines where they get an uninsured patient, and the hospital tells the doctors to hustle them back out of the door the minute the legal minimum obligation has been done? Healthcare cover equals access.
Worth tying in to this is the fact that out-of-pocket expenses are a major cause of financial hardship including bankruptcy, which insurance tends to mitigate. As above, these out of pocket costs are why people may be inclined to avoid seeking healthcare. Although the ACA may have been less effective than desired in this regard and medical debt is still a massive problem, studies have also suggested that the ACA has reduced medical debt problems to some degree.
Finally, this is also a major reason why the USA has such crummy overall healthcare outcomes on international comparisons despite ranking extremely highly on quality of care (if provided). Because in countries with forms of universal healthcare, where people are not going to be saddled with a massive fee for going to their GP and getting some tests, they're much more likely to go. Yes, their governments are "making people pay" via taxation... and they get better health outcomes for it. Hence also why even with the individual mandate scrapped, low income US citizens are still paying for their subsidised insurance: because they think they'll have better outcomes, too.
