My question remains: Why does the State of Alaska have an interest in what a (private) political Party does and/or who they nominate? What compels the State to fund a Primary (s)Election to benefit the minority of registered partisan voters?

My suggestions include, opening the Primary to all candidates (as perfectly described below by Lynn Willis) or charging the Parties the full costs incurred by the State or even getting rid of the Primary entirely and allowing the Party to nominate via caucus/straw poll similar to the Presidential nomination process