It's also massively more open, less controlled by parties, and more bound up in state law, than basically any other country's candidate pre-selection process that I'm aware of.
For a comparison, in this country, the process of parties choosing a candidate is entirely in the hands of party members - people who actually pay to be members - or sometimes even just elected party officials. Labor for instance have about 40,000 members which is an average of about 260 members per electoral district. Pre-selection of candidates is voted by those members. For the Liberal Party too, many candidates are chosen by votes numbering in the dozens.
Even then, quite often the choice is constrained by specific union arrangements and by deals between formalised Labor factions or by less formal factional tendencies in the Liberals. Member votes have regularly been overturned and head office candidates imposed, to meet factional balance objectives, get a high profile person into a winnable seat, achieve alternative action goals, remove unwanted candidates, etc.
In the Greens, I'm one of about 10000 paying members nationwide, so we have no more than a few hundred members in Canberra voting on all our candidate choices (everyone votes on all seats, not just the area we live in). Head office doesn't interfere in our case (there isn't really such a thing in the same way as the old parties) but there's a vetting committee to screen for candidate quality and alignment and a vetting committee report they send out with the ballots.
In all cases it's still entirely a matter for the voting members mediated by internal rules, a private affair for private party organisations, a small sliver of the general population. Certainly nothing like the effectively public votes the US parties undertake.
Worth nothing in many cases the US primary system is a tool of maintaining ballot access restrictions - the two dominant parties subject themselves to theoretically open primary rules via state law, which must be met to get on the actual election ballots, and then other parties failing to meet those rules can't be on the ballot. The "jungle primary" systems look like an example of this, usually pretty effectively restricting general election ballots to just the major parties.