@Thucydides - I don't claim to be an expert on the subject, but I've tried to follow the debates in the news and by talking with people who really know what is going on (public health professors). If anyone who is more closely involved knows more than this, they are probably right, but this should hopefully help at least give an introduction to the topic.
"What actually happens?" As I understand it, there are two major issues here. First is the huge cost of the uninsured. The process you described above is true only for people who do have insurance. Those who don't go through a different system. Hospitals (specifically ERs) are required to treat anyone who walks in, with or without insurance. As a result, if uninsured people ever reach a really bad state, they typically go to a hospital's ER to seek help. This creates two problems - at a smaller level, it puts incredible strain on the ER (it's much harder for them to keep down lines and still respond to real emergencies when the waiting room is full of Joe Flus who would have gone to their GP if they had insurance); at a larger level, it creates huge costs for individuals and taxpayers - once discharged, uninsured patients are billed directly for all the hospital's costs. These can be astronomically high, and probably unaffordable for someone who couldn't afford health insurance in the first place. If the individual can't pay, then the government, which mandates that the hospital treat everyone, is supposed to step in to keep the hospital from hemorrhaging cash, and the government ends up paying a lot (much more than it would pay for simple preventive care).
The second major issue of "what actually happens" is the insurance industry. Since it is a for-profit and largely unregulated industry, insurers do their best to make sure they don't lose money. This means that they often refuse to cover people because of (sometimes very minor) preexisting conditions, contributing to the uninsured problem above - for example, if Fred can't get insurance because he has diabetes, and then breaks his arm in a horrific Diplomacy negotiating accident, he will have to pay out of pocket to get his arm fixed, even though it had nothing to do with diabetes. This problem is compounded by the fact that the health industry isn't perfectly competitive - there are usually fewer than 10 insurance providers in any given state, and a few smaller states may only have a single provider option - so if that provider refuses to cover you, you're basically out of luck. Additionally, insurers will often try to wiggle their way around having to pay for treatments. For example, cancer treatment may be covered, but only as chemotherapy or surgery, rather than a more effective targeted therapy; or, your depression medications might be covered, but only for one prescription every four months, even though you need to buy it monthly. This effectively means that many people with insurance are only "partially insured," and this too contributes to the uninsured problem.
There are other issues with the current system, but as I understand it, this is the core part of the system that needs fixing. Insured or not, medical bills cause more than half of the bankruptcies in the US, and it is this problem, of rising and unaffordable costs, that the bill is trying to fix.
The bill - There are a few in consideration, but "the plan" that everyone is hyping right now is the Affordable Health Choices Act, the only one that really got moving in Congress before the recess. The Wikipedia page for it (http://en.wikipedia.org/wiki/Affordable_Health_Choices_Act) is surprisingly concis, especially compared to the Wikipedia pages on other health topics, but if you don't want to read it there, some of the main provisions, by category, are:
-Regulating the industry: Prohibiting exclusions for preexisting conditions, prohibiting variable costs (i.e., 26-year old Mike from LA pays the same as 26-year old Steve from LA, even though Mike's current plan actually charges him twice Steve's premiums for the same care because Mike's employer is much smaller), limiting the maximum out-of-pocket (deductible) amount a person can pay before the insurance company must step in, and establishing a Health Insurance Exchange which lets small providers or individual policyholders pool their coverage groups to reduce overall costs.
-Expanding coverage: taxing corporations that do not provide insurance and individuals who do not buy it, providing a health insurance tax credit for low income individuals to avoid punishing the poor through the above provision, incentivizing preventive care and long-term solutions (rather than temporary fixes) by reducing payments to hospitals with high readmission rates, and taxing people making over $350k to help pay for the plan.
Cost: The Congressional Budget Office puts it at about $100 billion per year, or about $25 billion per year in net costs when accounting for the taxes it builds in.
Hopefully that helps answer some of the basic questions of the insurance debate in a non-political way. Now you can throw in the politics and go nuts with the debate! Enjoy!