IMO the foundation has no business competing with other clients and features, be responsible for the core, let others build on it.
Nor does the Foundation have any relationship to Bitcoin Core.
I'm not sure I understand, this is from the foundation site:
As a non-political online money, Bitcoin is backed exclusively by code. This means thatultimatelyit is only as good as its software design. By funding the Bitcoin infrastructure, including a core development team, we can make Bitcoin more respected, trusted and useful to people worldwide.
Is this not a "relationship" to the core and its development?

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It doesn't (or at least, isn't supposed to) favour or control any given implementation.