IMO the bath, hair cut & trimmed nails is more for the destruction of evidence than out of care/concern for JAZ. If it was really care/concern, I don’t see that he would have been left naked.
Although DNA wasn’t yet on the radar, blood-type and hair/fiber analysis was (I believe). Perhaps JAZ scratched his killer, drawing blood, so he was cleaned up to eliminate this evidence.
We know police made great efforts to trace the box & blanket JAZ was found in, as well as the hat that was found nearby, so it is likely police would have also tried to trace any clothing he would have been re-dressed in - which could be why he was left naked. Also, clothes may have also contained the killer’s hair or hairs of any pets in the home which could possibly been traced back to the killer.
This makes me wonder if someone involved in the death may have ties to LE or at least been interested in LE/crime-solving.