In your case, $\bar x_{JK}$ and $\sigma_{JK}^2$ should be *exactly*
equal to $\bar x$ and $\sigma$.  You can show that by taking equations
(4) and (5) and plugging in the Jackknife definitions (and the
definition of $\bar x$) and doing some (perhaps ugly) algebra.  Of
course $\sigma_{JK}^2$ should become smaller with increasing $N$, but
so should $\sigma^2$.

In general the Jackknife results will be identical to the usual
average and variance as long as you analyze *linear* functions of
$x_i$.  It will only make a difference for non-linear functions.

At first and second glance I couldn't spot any error in your
equations, so maybe you're right and there's a bug in the
implementation.