The free market is pretty evidence based, so it would decide they are safe. I doubt the public insurance subsidy works the way you are representing it. They'd not insure against the risk of a catastrophic meltdown in a free market, the risk is too low. The "insurance" is probably more of a government tax on the assumption that there is latent risk. I'm not going to say that is unreasonable, but it doesn't have anything to do with how safe the industry thinks it is.
Consider Fukushima, the result of one of the biggest earthquakes in recorded history (Wikipedia says 4th largest [0]). Free markets don't bother to insure against that sort low-risk event. It would be like insuring San Francisco against the fact that it is on an active fault line - the market doesn't bother to insure against a risk that rare.
[0] https://en.wikipedia.org/wiki/Lists_of_earthquakes#Largest_e...
The free market is entirely based on investor hype. We have economic downturns ever 7-ish years, like clockwork, because some bubble burst again. It's a fallacy to treat investors like rational actors.
It's not like there are a variety of startups nipping at their heels. They want to make a profit but they're not doing it because they're convinced they will.
As for the word "current"... well, if the nuclear industry thinks it's so safe and thinks it can prove it to the insurance industry why don't they prove it and forgo their public insurance subsidy? They fought for this subsidy in the first place, they can give it up just as easily.
Why not let the free market decide if they're safe rather than providing taxpayers as a backstop?