^ The radiation hazard plays a big part. Sure, they could vent the reactor room (a process that would take several hours presumably, hence the need for Spock's sacrifice in the first place) so the engineers could enter (in their suits) safely and remove Spock's body. But then, Spock's body would still be emitting dangerous levels of radiation, necessitating additional precaution.
The way I see it, McCoy and/or and his staff entered the chamber in rad suits to prepare Spock's body and dress him in his burial robe. The engineers then entered the chamber with the casket (which would be radiation proof by nature) which Spock's body was placed inside, and then the casket was sealed shut. They then took the casket to the torpedo bay, confident that Spock's radioactive corpse would pose no danger in a radiation proof, airtight sealed casket.
Sure, they could have simply beamed him out, straight into the casket. But then you have to take into account the frequent complications with transporters and radiation, along with the fact that Spock was dressed in his robe. Again, they could have beamed him into the robe, then into the casket, by now it's become a comedy sketch! Starfleet would have procedures that emphasise respect for the deceased, when dealing with the preparation and burial of the deceased.
I am trying to keep this serious!