I think ideally the government sees it as net positive for democracy to maximize the participation of as much of its electorate as possible, in order for the process, and the winners, to have as much legitimacy and buy-in as possible. In an ideal world, no one would have to be "nagged", but realistically, everyone has limits on their time and attention span. Why is the college student who finally agrees to go with a friend any less of a valued participant than the retired citizen who has no other time commitments?

Party affiliation is part of voter registration in the states that run primaries. Primaries are not part of the Constitution, and before the 1970s, party candidates were picked via convention, i.e. party elites. Making the primary vote accessible to all voters was ostensibly an attempt to democratize the selection process. Seems like it'd be logistically difficult to hold party votes without having a record of voter affiliation.