There is no real reason for-profit healthcare would be more efficient at anything other than generating profit. In places with publicly funded health insurance, wait times for an appointment will depend on the urgency. So, you have to call a few weeks ahead if it’s a routine checkup, but you’ll be seen immediately without an appointment if you’re dying. Also, without the profit incentive, there isn’t much of a reason to schedule too many patients on a given day, so yeah, wait times are longer in for-profit healthcare countries.