Can't comment on anywhere else, but in the UK having insurance makes no difference to your treatment in the NHS. It means you can go to a private provider, but it makes no difference to your NHS provision.
I'm a Yank who spent nine years in Britain. After the first couple of years on the NHS, we gave up and went private for just about everything. As an example, my son needed three different operations. Each time we were told it was a six month wait on the NHS; we went private and were able to schedule the surgeries in 2, 3, and 6 weeks.
Technically you are correct that insurance doesn't change your treatment within the NHS. However, in several instances, we started with the NHS, only to switch to private at which we got better, faster service by the same doctors in the same hospital. So your statement is certainly misleading about the effect insurance has on health coverage in the UK.
In a sad anecdote, I watched our British neighbor go from a vigorous rambler to a shut-in after the NHS repeatedly brushed her off and mis-diagnosed her. Heaven help you if you're above 60 on the NHS. I'm convinced the NHS's philosophy is that all medical problems go away (from their perspective) if you wait long enough.