Sure you can find problems in any healthcare system. Because there is no upper limit on spending. There is always something more that you can do.
If you are proper rich* (dozens of millions or more) USA has the best health care system.
Fore everyone else, EU health care is better.
For instance right now I was at my pulmonologist and he wants me to take CT of my chest. As it is not emergency I could wait for a week (and it will be free), or I can have it today in private clinic, where I copaid 40 EUR*. If it were serious I would have it done for free today.
I got yearly prescription for Berudual, fosters, spiriva and Singular for free.
Bottom line is yes, sometime there is waiting time, but outside few pathological cases they are quite reasonable.
In USA I had more than double of my current pay, but with rent and my preexisting condition the way that they are, I am actually left with slightly more each month.
And there is another big difference between our system and US. Everyone is insured (weather it's public or private). So there is no fear of ever going to the doctors or calling ambulances.
Private clinics are getting paid by my insurance (I could get free CT at the same clinic a week later), 40EUR fee is just convenience (I don't want to wait) fee. And normally It wouldn't take a week, but with covid and flu season in full swing it takes a bit.