Why do Americans always dispute free healthcare by saying it will increase wait times when our wait times are already long?

My gallbladder has been acting up for almost three weeks now, I've had an ultrasound and a CT scan, and I need to have another scan next week to see if my gallbladder needs removed (spoiler alert, it does). After that, I've been told it'll be at least two more weeks to get into surgery. That's five weeks of walking to and from classes, spending hours puking in the bathroom every few days, avoiding all fatty foods (which, on a college budget, yeah right), waking up throughout the night with awful stomach and back pain, and pumping myself full of painkillers, NSAIDS, and nausea medicine just to function. I'm so sick of this. Just rip the stupid thing out already!

This whole thing was prompted by receiving the bills from the ER visit three weeks ago all the tests I've had since (there was lots of labwork in addition to the procedures mentioned above). I was complaining to my grandma and mom, who both had the same problem at my age, about both the wait time and the cost. Then my grandpa made the aforementioned comment that it would be much worse if I lived in a country with free healthcare. Ugh! I just want to tear my hair out with this whole mess!