You described exactly how the process could be changed: the way in which we largely elect officials in the US is First Past the Post, which intrinsically favors coalescing into monolithic entities. To change this, you must change
how we elect officials; the fracturing of the parties would follow. While it's not exactly as simple as that, it's probably the single largest step that can be taken. There are some other glaring issues that come to mind, though.
Unfortunately a lot of systems have been built up to largely favor maintaining power with the two major parties in the US. The third largest party, the Libertarian Party, still does not earn enough votes to earn federal funding. They also do not poll high enough to enter the national conversation through televised debates based on the rules set by a bipartisan (emphasis on bipartisan, not nonpartisan) committee.
The most successful Libertarian candidate has been Gary Johnson, who, coincidentally, went viral for his, "What's Aleppo?" response on mainstream media, and still was well off the threshold for federal funding. The most successful third party candidate in history was arguably Ross Perot, who was high profile, covered by the media, and was able to participate in the televised debates. He shot past the threshold for funding but ultimately the Reform Party has never been able to repeat the success they had with its founder, probably because they aren't billionaires that can bypass the enormous brick wall that is mainstream media.