it is entirely possible to implement elements of socialism within a capitalist market economy
Fixed it for you. Market economy ≠ capitalism.
You can have a market economy with social elements, but those elements are by definition not part of the ideology of capitalism and in fact in opposition to the goals and ideals of it.
whereby key parts of socially important institutions remain under state control while others are organized privately.
Yeah, that's the classic social democratic model. It's a compromise between the opposite ideologies socialism and capitalism rather than socialism mixed with capitalism.
One example of such an approach is the social market economy, which plays a role in Germany, Austria, and Switzerland.
It was developed among others by the economist Alfred Müller-Armack and seeks to “combine the principle of freedom in the market with that of social balance.” This approach, in turn, bears a certain resemblance to ordoliberalism (as advocated by Walter Eucken, among others), which had a formative influence on German-speaking countries in Europe after World War II.
Both of which arguably are forms of social democracy.
Unfortunately, however, this approach—which I consider sensible—is not immune to U.S.-style neoliberal capitalism (Chicago school of economics and others), which has, regrettably, also taken a firm hold in Germany today and whose effects are, among other things, significantly evident in the fact that the wealthiest segments of the population are becoming ever richer, while social benefits that benefit the general public are being cut back further and further.
That's EXACTLY the problem with social democracy in general, and also Neoliberalism in the US as opposition to downright fascism:
when you START from a position of compromise and the people on the opposite side of the table are people of wealth and power with no interest in compromising, you inevitably end up making unacceptable concessions.
ESPECIALLY when spin doctors and profit driven media get involved on the side of capital, which is the case in all countries that allow such antidemocratic practices.
This has been the case just about everywhere, including in my native Denmark where spin doctors were unheard of until the 2000 parliamentary election, and have been wreaking havoc on everything good about our politics ever since.