This is what horrifies a lot of us (British) about the NHS being further privatised. You create perverse incentives, beyond just providing the healthcare required.
Medicare isn't private. If it were, the premiums would be so high, that people would be forced to choose...
As I understand these things... Medicare is single payer and NHS is single payer & single provider. Which would make the British NHS more akin to USA's Veteran's Administration.
FWIW: I advocate single payer (Medicare for All), but not single (sole) provider.