How much would an airplane cost, if it were designed by engineers and NOT by the government?

Take a look at the price of an RV (completed) experimental vs the price
of an equivalent certified plane and you will get an idea. Basically
its about $80k vs $160k or thereabouts. There ARE some things about
factory planes that are "better" (arguable), like consistency of build
and support for parts. Factory aircraft also have a better safety
record. But the above price gives you an idea.

I know that Wipair quotes the STC for a set of $40k floats at about
$15k which is supposed to be the cost of the FAA certification. Of
course those are floats, not aircraft......

It's really hard to say exactly.....