Except that there is nothing nonsensical about a particle that has mass and doesn't participate in any SM interaction. It's inconvenient if such a particle exists, as it's very very hard to detect things precisely by their gravitational effects, but there is nothing nonsensical, or even particularly weird, about the idea. Plenty of particles only interact with a few of the SM forces - e.g. photons are not affected by the strong force, nor are electrons, neutrinos are affected by neither the strong force nor EM, only the weak force, gluons only interact with the strong force, not EM nor the weak force, etc