Every time some conservative or libertarian or anarchist or capitalist
claims privatization will help reduce the final cost and be more fair I
have to wonder if they understand basic arithmetic.

Sometimes they think they can pull the wool over our eyes by suggesting
that the inherent inefficiencies of government will be removed by a
competitive market place.

However I would argue the only really significant factor contributing to
inefficiency of government run programmes is corruption.

Corruption is indeed a very dangerous and real threat in a government of
the people, by the people, and for the people (especially if one or more
the "the people" parts get taken out by anyone wanting a bigger slice of
the pie than they deserve).  However claiming that corruption is more
costly than profit taking is where I lose it.

Oddly, or not, there's often far more corruption in a free-for-all
capitalist marketplace, which only adds to the costs along with profit
taking, which brings us back to having to regulate things from a
government level (which is never 100% effective), which brings me back
to asking how, exactly, is profit taking saving society, i.e. the 99% of
society and especially the lower 50% of it, any money?

The only money that can be saved in a fair economy is the excess profit
being taken by the 1%.