Yes, private industry will fix problems by choice rather than being required to. When seatbelts first came out, all manufacturers offered them as options rather than standard until Saab and Volvo eventually made them standard. Ford offered them as options for about a decade before making them standard. Even once they became standard (usually required by law), most people didn't wear them until they were required to by law. Requiring cars to have seatbelts and people to wear them has saved millions of lives. Car companies would not and could not have accomplished that.
For another example, Boeing knew that its 737-MAX had problems with its software and hardware. It offered additional safety features that would have prevented the two crashes that killed ≈ 350 people. After the first crash, did Boeing recall the planes to have them fixed to ensure there would be no further crashes? No, they lobbied governments not to ground them. After the second crash, did Boeing ground the planes? Again, no, they worked hard lobbying the US FAA and other agencies not to ground them. The only airlines who grounded their 737-MAX fleets were the ones who'd had crashes. Others largely kept their planes flying.
The country with the most significant regulatory capture by Boeing took the longest to ground the planes because more libertarian-minded Republican officials were happy to keep it flying until pressure became too great. Regulatory capture of government by industries does untold damage by allowing private companies effectively to self-regulate.
If private industry is so ethical and responsible, why do companies sell lead paint wherever it isn't banned? We've known lead is harmful for more than a century. Why do fossil fuel companies stick with producing fossil fuels that they've known for decades (because they funded the research!) cause climate change?
It is demonstrably false that private industry would be more effective than government at protecting lives or serving the public good. Governments aren't perfect, but you'd have to live in La La Land (aka Gault's Gulch) to believe that private industry is superior.
