Billionaires tend to be far more critical of what their money finances than government granting authorities.
True, but the outcome is not usually what you are implying. Billionaires tend to put their money where there is the most to gain for themselves, while governments have a stronger motivation to fund important fundamental discoveries that do not provide an immediate return on investment.
Consider all of the scandals involving made up data.
Both privately and publicly funded entities do this. At least publicly funded entities can be cross-checked. Privately funded entities are under no pressure to disclose all their sources, and will be even less so as private funding of science becomes more socially acceptable.
A billionaire who discovers shenanigans certainly won't fund that researcher again, a government agency probably will.
To a billionaire, "shenanigans" means that the "researcher" didn't arrive at the results the billionaire paid for. So yes, the billionaire will not fund that researcher again.
...it's pretty obvious that private donors are more likely to scrutinize than public sector donors.
Yes, but only to make sure that the private donors' political biases take precedence over the truth.
Billionaires have the luxury of blowing their money however they see fit.
And they will only "blow" their money on endeavors that make them more money. How do you think they became billionaires to begin with?
This is how science got funded during its first centuries as a discipline when many of the giants of science did their work.
Lots and lots and lots of good science had to fight and uphill battle against the political desires of private patrons back then, which held back scientific progress rather than promoted it.
No, private funding of the sciences was, is, and will be a disaster.