[Pervinca Took]

Alfirin made the point that the splinter was now very near Frodo's heart. Or, at least, "deeply buried," Gandalf said. It tends to be assumed that it was working towards his heart, because Gandalf said that was where the Witch-King was aiming for. Perhaps it was simply known to the Wise that that was how Morgul-blades operated, how they turned a person into a wraith.

Gandalf believed all along that there was a fragment of the blade still in the closed wound, and Strider and the hobbits had seen that the tip of the blade was broken off. I presume that Aragorn could have tried extraction in the wild simply with a knife, yet he dared not. Maybe Morgul splinters had a nasty habit of resplintering - but if so, why was only one splinter found? Maybe they were simply elusive and only Elrond's skill - and the presence of Vilya at Rivendell - were enough to give Frodo a fighting chance of surviving extraction even if it could be achieved. At any rate, by the time they reached Rivendell, it must have been too late for a "scapula" method - it has travelled too far from the original site of the wound.

The pain travelled both down his (Frodo's) arm and his side - or, at least, the chill did, as did the sensation of "icy claws" - claws implies pain as well as cold. That makes sense in the early stages, but even later on, when the splinter is supposedly near his heart, he still cannot raise or use his hand. Reasonable enough - it was a deadly wound with residual damage. Then again, when the pain recurs even after the healing, it appears to be at the site of the initial wounding, not where the splinter ended up.

Further to Jallanite's point that the surgery could be "local" once the splinter was located, Glorfindel's searching of the wound with his fingers (by which he learns enough to disquiet him) might be the key to the "locating" aspect. It takes about three days, too - even a master of healing would need to know exactly where he was cutting.