I live in a country with universal healthcare (actually a dual system with universal healthcare alongside a private system) and I've rarely been in a medical setting where I've seen people being genuinely entitled and demanding for no other reason than "I don't have to pay, so gimme". Of course if you're in the ED you'll always get those folks that start getting shirty if they've been waiting a long time, or the frequent flyers that are there for other reasons deciding to throw a fit, but the only time I've ever seen people basically waltz in and demand to be seated, oops I mean demand to be seen in 5 seconds flat were folks who pulled the "I pay good money for my private health insurance, so you will see me right now" song and dance routine.
The main way I, and most other people I know, utilise a universal healthcare system is with stuff like making sure we get regular health checks, and visiting a GP when we're unwell. Not having to pay out of pocket, or at least not having to put ourselves in financial strife to get healthcare does mean we're probably more likely to go to the Doctor when we need, but I don't think that's the same as demanding everything including the kitchen sink just because we don't have to pay for it upfront.
I think for me one of the perfect example between attitudes towards healthcare here (Australia) and the US came when I was diagnosed with marginal zone lymphoma last year (thanks to those regular health checks I mentioned prior). The cancer was staged at 4a, so not currently symptomatic, don't need treatment at this time & I'm under a 'watch and wait' protocol. The only section of my friend group who were aghast at the 'no treatment necessary at this time' deal, and who emailed and messaged to let me know that wasn't good enough, and I should start kicking up a giant fuss and demanding treatment, were some of my American friends. And yes some of them did suggest that the fact I wouldn't have to pay out pocket should mean there was nothing to stop me demanding everything be done right this very second, and it was like, "Okay, but why would I do that when it's not medically indicated?" They honestly just didn't seem to grasp (at least not without a lot of explaining and reassurances) that I would hear the word 'cancer', hear the words, 'treatment not needed at this stage', and then just be accepting of that.