We have a similar system in California called the jungle primary—basically there are no party specific primaries (except for president because this system is incompatible with other state’s elections), and the top two advance to the general election.

There are a few issues though. If a candidate wins more than 50% of all votes in the primary, they win the election and don’t appear on the ballot in the general election along with the president. Since there is generally higher turnout for the general election rather than the primary, you can sometimes have a generally unpopular candidate win in the primary with 50+% of the small number of primary voters.

We also have issues with spoilers—if a bunch of similar candidates run, and all split the votes between them, it’s possible they don’t make the final ballot, even if any of them individually would have won the final election. This seems like a fringe issue until you realize that parties have actually supported lots of minor candidates on the opposing side in order to eliminate an otherwise dangerous challenger.

So overall it is somewhat better than first past the post but it still has significant issues. In general I think elections that select a single candidate are somewhat undemocratic by nature and we should think about ways to give the minority a voice but not the ability to shut things down. This may be a difficult balance to achieve but it’s still worth aiming for.