Not to defend anything about the American model (I personally hate it and think it's broken) but the annual checkup is a very cheap way of dealing with a possibly expensive condition.
The checkup is non-intrusive and for presumably healthy people consists of a physical, basic blood tests (glucose and some metabolic indicators + infection markers) and a blood pressure check. As an example, it's way cheaper to find out you have pre-diabetes this way, than to go to the doctor with symptoms when you have full on diabetes. This is mainly the reason why insurances pay for these in full, as they're part of 'preventive care'.
> How much more would it cost our health-care system, if everyone took advantage of
Annual checkups are definitely not the reason healthcare costs are sky-high in the US.