There is no such thing as the DNC's primary. The only Presidential nominating election run by the DNC is at the convention, when they delegates vote on the nominee.
State parties in some states hold elections in the nominating process in the form of caucuses. Primaries, where they are used, are run by state governments, not the DNC or state Democratic parties.
> If the DNC would just "appoint" its favorite candidate, that would be fine.
That's actually what the convention is: its where the DNC appoints its favorite candidate.
> It's kind of a normal thing to do in Europe, actually. But then, just like in Europe, the Dem and Rep parties would have to allow other parties in the general election as well, and set up a system that works for third parties, because chances are more often than not people wouldn't be happy with the main parties' "appointed candidates".
No, they wouldn't, and even when there was no pretense that the nominating conventions were anything other than insider-driven affairs (which only ended fairly recently with reforms in both parties to rest more heavily on either state-run primaries or relatively-open -- compared to how they used to be -- party caucuses to select delegates, or at least inform delegate selection) the electoral system did not support third parties any better than it does now.
The GOP already did this to Trump at one of their debates[1], and no one threw a big fit about it then. Not a Trump supporter or claiming that it's justified, just saying it's not unheard of.
[1] http://www.vox.com/2016/2/13/10987776/republican-debate-audi...
If the DNC would just "appoint" its favorite candidate, that would be fine. It's kind of a normal thing to do in Europe, actually. But then, just like in Europe, the Dem and Rep parties would have to allow other parties in the general election as well, and set up a system that works for third parties, because chances are more often than not people wouldn't be happy with the main parties' "appointed candidates".
Both the DNC and the RNC want to maintain the pretense of "fair elections" within their parties, so that people think their voices are heard, so they don't need to join another party and form a competitive grassroots movement.