"I would agree with you in theory, but in practice, most people who don't have insurance that covers office visits, prescriptions, and routine checkups just don't ever get them."
It's hard to argue with an uncited generality, so I'll accept your statement for the sake of argument. In any case, I look at that as more of a case that people are less likely to indulge in a doctor's visit if they're responsible for the bill (or rather, are aware of what they're actually paying for it). When people live under the illusion that they're only paying a co-pay for doctor's visits (after all, employer-provided health insurance essentially is paid for out of employees' paychecks), they're more likely to go to the doctor every time they get the sniffles. More visits = more health care expenditures, irrespective of any hypothetical "sicker due to going to the doctor less" population.
"Limiting what insurance companies have to pay for to catastrophic/chronic trauma/illness would in practice result in more of those expensive sick people, and people using the expensive ER as a clinic because they don't have the money to see a doctor - exactly what we have now."
Only if people are as uninclined as you say to avoid doctor visits and the like if insurance doesn't pay for it. I think if people came to realize how much cheaper it can be to pay for non-catastrophic care out of pocket many if not most would be inclined to do so.