But there was more time spent playing with pretend bandages, and giving pretend medicines, working out how to apply a sling and putting on sticky plasters than any sexual exploration.
I think this is a cultural difference. In the US, 'playing doctor' is used much more narrowly, to specifically describe the action of young children taking their clothes off in front of each other and/or 'examining' the areas of the body that are usually hidden by clothing.
Someone in the US can correct me if I'm wrong - but the common result of playing doctor is seeing and/or physically interacting with the genitalia of someone near your own age and/or opposite sex for the first time. The curiosity is rooted in how those opposite parts work, OR what it's like to see a same part from the outside, as an observer.
So I think here in america, it's pretty much always been the name of a game that is somewhat taboo (because it involves hidden sex organs) but common (because children at a certain age become fascinated with their genitalia - usually they've recently learned the word for it, and that combined with the onset of wearing clothes as a requirement - covering up - turns it all into something kids hide from parents but can't avoid being curious about).
So, that's how a normal part of early human development - the discovery of the sex organs - turns into a 'game' that has to be called by a euphemism, because talking about sex is scary/taboo/not for common conversation/whatever.
A little bit of research is showing me that yes, in the UK 'doctors and nurses' involves actually doing things with stethoscopes.