This does not work. People live in rural areas because they don't want density. That is the whole point. (Alternatively, it might be because they are too poor to move. However, densifying/gentrifying an area does not fix that--it just boots people to a different poor rural area.)
You are trying to make rural areas viable by making them a city. You might as well just improve the situation in an already existing city.
In Europe, a lot of the small towns are pretty dense, due to the historical need to preserve agricultural land and have a community to rely on[0]. In ex-soviet countries, you will often see in the countryside condominiums, as it was then seen as much more efficient and better community-wise.
So yes, if you densify rural towns to reduce sprawl, it becomes much easier to provide good services like doctors, groceries and schools.
[0] Here is one that comes to mind: https://en.wikipedia.org/wiki/Cordes-sur-Ciel