The phrase "a core responsibility of" is correct and usable in written English.
You can use it to describe an essential duty or obligation associated with a particular role or position.
Example: "Managing the budget is a core responsibility of the finance department."
Alternatives: "a fundamental duty of" or "a key obligation of".
Oversight is a core responsibility of my job as a member of the Senate.
Of course, plenty of people don't think that guaranteeing affordable health insurance is a core responsibility of government.
The question is whether it represents the future of healthcare, which traditionally, has been a core responsibility of national governments.
Maintaining peace and security is a core responsibility of the United Nations, and any allegation of wrongdoing by peacekeepers is taken extremely seriously.
Regulation of immigration and borders is, of course, a core responsibility of the executive under our system of separation of powers.
Transport is a core responsibility of London's mayors and bold transport policies would be at the core of a Wolmar mayoralty.
Union leaders, like political leaders, are charged with a core responsibility that isn't negotiable: the task of leading.
But I think the main thing is to have a firm belief about the core responsibility of solving these challenges, and not delegating that to someone else.
If the Great Recession taught County leaders anything, it is that our ability to meet our core responsibility of providing a safety net for those in great need -- the elderly, the disabled, the homeless, the youth -- while ensuring that our communities remain safe and healthy, is intimately and extrinsically tied to the health and vitality of our economy with the tax revenues it affords us.
This year, voting is more than just the core responsibility of citizenship; it is an act of defiance against malicious political forces determined to reduce access to democracy.
The core responsibility of getting the best women players in the world out on the course playing for money and on TV is a fairly simple business challenge.
