I'm sympathetic to the concept, but I think that the advantages that organized parties have in terms of coordination (e.g. people with broadly similar values and policy goals choosing one candidate to represent those goals to avoid splitting the vote and seeing someone antithetical to those shared goals elected) are sufficiently strong that you would just see the current primaries replaced immediately by a primary process run completely independent of government oversight and resources. I can't imagine that being good from a perspective of electoral legitimacy or reducing the influence of money in politics.