Earlier quoted context omitted.
I have a "radical" idea: why not actually let the constituency of this party make the choice.
Or actually radical: switch from our terrible first-past-the-post voting system to - say - ranked choice (or one of many alternatives; they're almost all better than fptp) and then primaries won't be so important and parties won't have so much power over our kinda-democratic-but-actually-oligarchic political system.
I’m not sure why they would reduce party influence either. Features like being robust against spoilers would seem to most benefit major party candidates.