This article mainly covers the MIT fusion effort. For years they struggled to keep their government funding, despite their tokamak having the most powerful magnetic field of any in the world. They finally lost that battle, and now private investors are stepping in. It doesn't appear that there was an alternative.
Government can obviously throw more money but it tends to be more conservative about what it funds. And private funding can go pretty far; it was interesting to see that TAE (Tri Alpha Energy, an aneutronic fusion effort) is up to $800 million in funding. They started work in 1999, so long-term fundamental research does seem to be achievable in a startup setting.
Even NASA in the peacetime 1950-60s which was so famously efficient and effective was largely the result of talent quickly absorbed from private industry and academia into one organization. It was a newish organization which held many of the benefits of non-gov/private organizations, as they were not yet fully affected by the heavy bureaucratic and political load which hits every government agency over time.
The type of stuff that scares away the raw talent and creatives and shifts to a system which values people who play politics and shifts power to administrative roles over the producers. https://www.jerrypournelle.com/reports/jerryp/iron.html
The more difficult question: how would a startup come to experiment with ways to build a self-sustaining fission reaction if the basic concept hadn't been demonstrated before? Something would have to be very different about history. As an improbable hand-wave, let's say that academic scientists had only ever experimented with fast-neutron fission, and didn't realize that natural uranium could sustain a chain reaction if the neutrons were moderated. We're probably deep into alt-history "World War II never happened, also the world has been surprisingly peaceful" fiction with that hand-wave.
Finally, if you really mean current nuclear power technology, e.g. one of the actual Generation III reactor designs currently operating, plus supporting infrastructure -- no, a startup could not invent all of that.
https://en.wikipedia.org/wiki/Generation_III_reactor#Generat...
You could easily blow through 100 million dollars just trying to prototype the enrichment process for the uranium that your reactor needs as fuel. Never mind actually sourcing enough natural uranium in a world where uranium has previously been a niche material like rubidium, building a uranium enrichment plant, fabricating fuel elements, forging a huge pressure vessel, developing the whole field of nuclear engineering...
I don't think that a reactor requiring enriched fuel is a good Very First Reactor design -- and the actual first reactors did not require enriched fuel -- but enrichment came up in the course of answering whether a startup could invent a modern reactor in the absence of an existing nuclear industry.
Molten salt reactors, for example, were known back in the 1950s, and appear to have a number of major cost advantages.
Or you could start a MSR with U-235 enriched from natural uranium, but that would require developing complex and expensive enrichment technology before you get your first watt of nuclear power generation.
I am genuinely curious if for-profit startups is the right setting for "fundamental breakthroughs". Could the current nuclear power technology have been invented by a startup for example?