If a party wants to hold a closed primary, can't they do that themselves without help from the government? Why would the government be involved in a party-internal election?
As to "why" the government should feel obligated to subsidize the process -- because the government has the ostensible goal of facilitating fair and proper elections, and presumably the primary process -- which is not Constitutionally-enshrined -- is a net benefit to the general election, at least compared to selection-by-party-convention. In the future, political parties may decide that it's better to have open primaries, but that's orthogonal to the government providing the voting infrastructure and logistics.