My confusion comes because in my first two citations you suggest requiring patients to pay for most of their basic/non-life-threatening healthcare. You say this, if I'm not mistaken, because "free or semi free" healthcare is "operated as a cost center". That is, by requiring patients to pay for most of their basic healthcare, that cost center will be reduced because patients will be discouraged from seeking "unnecessary" healthcare.
On the other hand, in Denmark where basic healthcare is free, the data you allude to (but do not present) show that even then "a lot of people don't go for checkups [because] it's not a common thing to do." Additionally, you write "Furthermore the situation is even worse for men over 50."
So my confusion comes from your contradictory assertions about the effects of fully subsidized basic healthcare. Either free healthcare generates cost overruns because people seek out unnecessary treatment, or free healthcare does not generate cost overruns because people do not seek out even necessary treatment.
Your assertions about the effect of free healthcare are paradoxical, hence my confusion.
EDIT: fix typo, recast first sentence of penultimate paragraph. Remove redundant prepositional phrase from last sentence.