I'm not sure about the US system but here ER system is ran based on the urgency not based on who came first. Many people come to the ER with not so urgent issues and then complain when they wait for a long time because there's (a shocker I know) people with actual urgent issues that need immediate attention. If it's empty that day you will be admitted for those non urgent issues immediately. I remember breaking my arm, going to the ER and waiting for two hours because there were traffic accident patients and open fractures that had priority over me.

Some specialist-provided tests have long wait lines, but everything depends whether it's urgent or not. For example my bro waited on an elbow MRI for 1,5 years. He had minor pain when lifting weights but otherwise no issues - this wasn't urgent and the waiting time was not unreasonable. Sometimes people wait on surgeries for a long time but these also aren't urgent otherwise they'd be done very soon (like hip breaks, cancer removals etc. are done).

Croatian media and laypeople (I'm a nurse and soon to be an MD so I have some work experience) tend to misinterpret the waiting times and make it look like people with brain tumors are waiting 1 year on their MRI or breast cancer patients who are waiting 2 years on their ultrasound. This is absolutely not true and if you're admitted to the hospital with say an acute gall bladder inflammation you will be ultrasouned the same or the next day. Some of those services with long wait times can be done in private practices and they are not expensive compared to the US prices, even when you adjust for our lower standard of living and wages. The only exceptions are surgeries, especially cosmetic and ortho surgeries. We even have private labs where you just walk in, tell them what blood tests you want done and pay for them right there with zero waiting time (excluding possible prep like fasting the day before if needed).

The system sucks sometimes and there are different issues that patients and providers face, but everything is avaliable if there's really a need for it. I have my issues with it and might be going abroad to work, but at least I know I won't die because I can't afford a yearly checkup or a live saving surgery.