Can you back that statement up with some data? In Switzerland, they mandated insurance coverage for everyone thinking the same thing. Everyone ended up using it more for the smallest of things, running costs up even further.
They may have been using the system more as you state but there is no data that I can find saying it cost more overall.
An example- With no insurance, Jane Sixpack has some pain and goes to the ER (b/c no doctors office or clinic will see her without money up front or insurance). She has labwork that shows she has an ordinary UTI. However, because she came to the ER and the physician will be liable for anything else the patient may have, the physician orders an ultrasound to rule out serious problems. She has bloodwork. She spends 3 hours in the ER. Total bill- $2000-3000.
Same patient another scenario. Did not go to ER because she has no insurance. Simple UTI spreads to her kidneys. Now seriously sick. Calls ambulance. Brought to ER, admitted to hospital, receives 7 days of IV antibiotics in the hospital. Total cost-- $10000
Now --universal care with over utilization by Jane Sixpack. Goes to doctors office 10 times in a year b/c she remembers pain from last kidney infection which required her above hospitalization. Hypochondiac never has a UTI. Ten urine cultures done and bill for 10 office visits. Cost $1000-2000
Even with overutilization as you claim will occur, it would still be cheaper because of access to care early in disease course and care can be initiated at lower expense. If you don't believe me, go sit in anER waiting room for a day and see what you are paying for. At least half of the patients there do not have anything remotely resembling an emergency. Just getting a doctors visit at the most expensive and inefficient way.