Can a finite massive object enter the horizon in finite external coordinate time?

Textbooks say that for a distant external observer, an infalling object never reaches the horizon because $dr/dt$ tends to 0 as $r$ approaches $2M$ . But this conclusion assumes the object is a test particle with negligible mass. If we take the object's own mass $m$ seriously, then the exterior spacetime is determined by the total mass $M+m$ . The new horizon is at $2(M+m)$ , so the old surface $2M$ is no longer the true coordinate singularity. That means the object's radial velocity does NOT vanish at $2M$ ; it only vanishes asymptotically at the new horizon $2(M+m)$ . So the coordinate time needed for the object to cross the original $2M$ surface is finite. My question: Does this show that the "eternal freezing" result is only valid for test particles, and that a finite massive object can indeed enter the original horizon within finite external time? Am I missing something?