> People can't reliably see dentists or get doctor's appointments because the nationalized health system can't manage capacity properly.

Not accurate in the slightest, this is due to the current administration deliberately funding the NHS less than it needs to function effectively.

> People can't reliably move around the country because the public sector transport workers are constantly going on strike

Only currently true because of similar reasons, it's not structurally true of that sector.

> Putting the public sector in charge ... And now the UK suffers massive healthcare shortages. Same problem.

Not accurate at all.