No, non-partisan blanket primaries make things worse; you end up with the same bias toward the hypercommitted ideologues in general election candidate selection (primary voting) as with the normal primary system, with the added issue that if closely-related factions are overrepresented enough in the pool of hypercommitted ideologues in a region, only candidates from those factions (which are often indistinguishable to anyone other than hypercommitted ideologues) end up available for voting in the general election.

The real solution isn't to change how we select candidates for what amounts to a "choose one of two options" general election, the solution is abandoning election methods which make the general election into a "choose one of two options" elections.

For legislative elections, replacing single-member districts with 3-5 member multimember districts and using a candidate-centered (not party list) election method using preference ballots that tends to produce proportional results (Single Transferrable Vote or something like it would be the obvious choice) would be ideal.

For state-level executive elections where there is a constitutional designated successor (e.g., Governor and Lt. Governor), electing both from the same ballot, using a preference voting system -- pretty much any sane single-winner preference voting system, including Instant Runoff, though a Condorcet method would probably be preferable -- where the normal winner is elected to the top slot and the winner from the same ballots with the first winner eliminated is elected to the second spot would be an improvement (preference voting minimizes the incentive to vote tactically among parties, having two offices to fill from the same ballot encourages each party to provide more than one candidate.)

For elections that must be single winner without a designated successor (such as, I dunno, State Insurance Commissioner), using the same single-winner preference-ballot system as for executive with designated successor -- in its straight, one winner form -- is probably the best available choice. OTOH, as these are often executive officers, it may be better (with a decent election system for chief executives) to eliminate separately-elected executive officers so as to minimize split attention and focus accountability.

For all elections, using partisan primaries but using a preference ballot system that selects the appropriate number of candidates for the election (rather than FPTP) would be the best choice. While you'll inevitably always have a bias toward the more engaged in non-general election, preference ballots minimize the influence of the most pure faction within the most engaged subgroup that votes in primaries, and fixing choice problems in the general election mean that there are more parties whose primaries matter.

Presidential elections, assuming that the electoral college is retained -- and that's the hardest change to make since it requires a Constitutional change, while most of the others do not -- are the hardest fix. Changing to preference ballots with a sane single winner system for primaries would help choice issues, but would probably end up turning actual candidate selection back over to party insiders at conventions, since it would make it much harder for a single candidate to get a majority of delegates through the primary process.

Changing primaries to preference ballots and using a system that outputs a complete preference ranking of candidates by state -- such as iterating a single-winner system eliminating the winner after each round until a complete preference ordering is achieved -- and having convention delegates vote preference ballots (that are, in the initial ballot, required, for regular state delegates, to be cast in accordance with the outcome of the state primary on the first ballot) would be one alternative which would both mitigate the power of extremist factions and preserve popular rather than insider selection.