But that's exactly what a union is: a lobbying group that strives for a better outcome on behalf of its members. "Independence" is not part of their job description. Why wouldn't they talk to politicians, or even publicly support the political parties that are more aligned with their goals? Or do you believe that is a privilege to be reserved only for the rich?
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Sure be a lobbying group but without public funding. If your members can't be bothered to pay the cost of the union it can cease to exist.