Swiss people are perfectly capable of becoming health workers? What kind of argument is this?
Fertility rates are low and people are ageing, like everywhere in Europe. There will be a moment that simply there won't be enough workers. The reality is that there is already a lack of healthcare professional even without a population cap that would only get worse given the case.
That's an option, but it takes a long time to train and recruit locally, costs a lot of money, and you'll probably have to increase salaries to get the required numbers. If there were an easy and cheap way to recruit all the required staff locally, that would already be happening.