I think the point was that it's misleading to say "Well in my home country, it's free (or super cheap)" because that's simply not true.
if similar amounts of tax go into healthcare, and then you pay that same amount again on insurance, then that makes other countries about twice as cheap, which is just super.
And, what about the fact that when you don't pay tax, you can still get a doctor to remove your spleen? That's the "free" part. Lose your job: none of this panic.
This past financial year, I've been working on my own project with a tiny bit of contracting to get by, which means I'm way under the tax threshold of $18k, so I pay zero tax on my income. When i go shopping i of corse still pay GST(VAT) and other minor public revenue streams, but I can still get a free general practitioner, or a blood test, or help in paying for a specialist like a shrink as long as a GP writes a referral, and if I need complex surgery I can go on a public waiting list, or if I'm injured I can go to the ER and be X-Rayed ands treated.
By paying through tax, which is within everyone's means (no income like me right now = no means so no tax, or later if I make millions I'll be earning way beyond my own needs so I'll pay higher tax), yes, some people do get free healthcare and everyone has a doctor when they need one.
Trying to attack free healthcare on that ground is like trying to argue that public roads and toll roads are the same, when they're obviously very different. You don't need a tax payment receipt to get on to a public road, despite it being created through taxes.
There's also big differences in where your money goes. Excessive taxation goes to a not-for-profit service provider with a motivation to do well by their customers. Excessive insurance payments go to some rich shareholder with a motivation to take as much money as they can from you.
Also, in a lot of these countries, foreign tourists get the free healthcare too, despite obviously having payed no tax.
Meanwhile, a friend of mine that went to the US got a bad flu and went to a doctor, who recommended a flu vaccine while he was there. He was billed over $1k for being uninsured! At a private doctor here (not part of the free system) you'd expect the shot at near cost price ($20-$100) and the consultation to be around $60. That's a super big difference.