It's not just the ratio. In many medical roles, engaging your full humanity with every patient would destroy you psychologically, even at a much lower number of patients.
"Follow the process, follow the training" is how medics, emergency responders and the armed forces are able to stay in the job more than a few years without burning out completely.
(It's also, as psychological defensive mechanisms go, somewhat fairer than those used in the past. Ask a retired medic in their 80s or 90s if you know any.)
Healthcare professionals know this to be true. This is why when their own loved ones are the patients they have such a strong tendency to become very actively involved —- it’s not necessarily that the person attending to their loved one is incompetent, but chances are that their loved ones will similarly be just another face that occupies another physician’s mind for a few minutes.
Artificially high barriers of entry in the field may lead to massive compensations but also to a huge ratio of patients to physicians — this takes a toll.