It's true. Dwarf planets do have to be round ("in hydrostatic equilibrium"). That requirement will definitely disqualify many of the hypothetical billions of Oort cloud objects.

Ceres is an official dwarf planet, but I excluded it from my list because I was focusing on trans-Neptunians.

A big reason why the IAU hasn't confirmed most of the 36 is because they want strong evidence of roundness, like a spacecraft flyby with direct imaging. Pluto and Eris are close enough that earth telescopes can just barely resolve some of their shapes.

Finally, Pluto and Charon are too close to be considered in the Oort cloud. Sedna, whose discovery precipitated a lot of this crisis, has been nominated as the very first discovered object in the "inner Oort cloud." Sedna's perihelion is at 76 AU.