I understand the formula for the Boltzmann distribution to be
$P(E_i) = e^{-E_i/(kT)}/Z$
When the energy levels vary continuously illustrations of pdf for either the energy or the velocity at a fixed temperature show a skewed bell shaped distribution over the non-negative reals somewhat like a smooth version of the Poisson distribution. but the above formula is a decaying exponential wrt the energy level and everything else is constant at fixed temperature, so I would expect the graph of the pdf of energy to be monotonic decreasing & convex. The pdf of the speed would then to be the the right half of the Gaussian distribution which is still monotonic decreasing. Either way I do not see how the pdf could be increasing near 0. Clearly I am missing something really obvious, but I cannot figure out what it is.
What you are missing is the density of energy levels. Your $P(E_i)$ is the probability for a single state of energy $E_i$ (in a discrete situation). But the number of such states in an interval $(E, E + \Delta E)$ will not be constant. For example, for a single particle (with no internal degrees of freedom, just kinetic energy) in a box, this number is approximately proportional to $E^{1/2}\ \Delta E$.