I'm not interested in the politics, but I can perhaps give some input from someone who has actually experienced another system (and not just on holidays). Lived the first 30 or so years of my life in Switzerland. Had our first baby there, and the second here in the UK. Had few other experiences with the NHS. Get private cover (Bupa) through work and used that a couple times too.
In Switzerland, health insurance is mandatory by law, but it's private and there are many providers who have to cover what's in the law, so prices end up similar (competition). Everyone pays the same regardless of salary (great for top earners), and there are state subsidies if you're on low/no income. For example, I used to pay around 250CHF/month (about £220) for the most basic cover, and that had a 2000/yr excess (which I never reached), so effectively any visit to the doctor I had to pay for out of pocket. You can pay more to have a lower excess, and also pay extra to have things like private rooms (should you be hospitalised), and cover for "alternative" treatments like physio etc. Another example, my parents pay over £400 per month, each (they're retired btw). Kids "only" cost around £60-70. All these costs vary by canton - so you will pay significantly more in Geneva than somewhere in the sticks. To put these numbers into perspective, someone working an average job might be making 4-5k CHF gross per month, so to support a family of 4 you're looking at easily over 10% of that going to health insurance! If both parents are higher earners they would pay the same net amount, so a much lower % of their income (unfair imo).
These healthcare costs are the nr 2 reason for people to go into debt, after taxes. (Which are also very different but that's for another day!) The way the excess works means most people pay a lot but get nothing. Glasses & dentist are not covered anyhow.
The way the system is setup creates a tug of war between the hospitals and insurers: the hospitals usually want to turn a profit and tend to overbill, whereas the insurers are always fighting to not pay (as they do!), which has created a very complex billing system that everyone hates (Tarmed). My doctor friend tells me that the hospitals employ full-time people to add extra stuff to the bills, and these positions pay for themselves! Unsurprisingly, both the hospitals/clinics as well as the insurance companies all make massive profits (and they for the most part privately owned). As a result the premiums rise every year and people complain a lot about this.
Overall the system creates an incentive to overdiagnose and overtreat, in stark contrast to our experience here in the UK where the incentive is to pragmatically control costs. For example: during pregnancy, there is a gynecologist appointment every month, in his private practice with his ultrasound machine. Nice to see baby every month, but costly! Here, we had the 3 & 5 month checkups done by a midwife/nurse, and things only got involved in case of complications.
Our biggest and perhaps only gripe here is that it's very hard to see a GP, as the village practice is way oversubscribed. You have to spend hours on the phone and they are generally technologically-impaired. Private insurance in comparison I can get a 10-minute video call with a GP the same or next day via their booking system/app.
Comparing our 2 pregnancies & births ... Overall we liked the UK one better I must say! This is partly down to the culture here being much more human & compassionate, which counts for a lot in those moments. The swiss doctor who did the C section for nr1 was beyond rude and showed negative human skills during the operation, shocking! The same operation at the Rosie in Cambridge was much nicer with the staff even singing at the end

During pregnancy the more hands-off approach didn't bother us, although maybe that's because it was the 2nd time round.
Anyway, long story short, it's not always greener on the other side. Incentives matter a lot.
Edit: oh crap that's a wall of text ... Sorry!