From the very beginning of book 24, this seems less like a heartfelt supplication
than some kind of Olympian
conspiracy to stop a fued between gods. Zeus
wants to quash the conflict between Apollo
(Lines 65-75) and devises a plan whereby both of their beloved mortals, Achilles
can maintain their glory. This doesn't start off on the same footing as, say, the supplication of Agamemnon
or of the supplications between the gods.
It seems more of a supplication between all the gods, Hera being their representative, (line 98) and Thetis, with them trying to induce thetis into convincing Achilles to give Hektor's body back to Priam, this request goes from The Gods to Achilles and Priam, who play out this last scene as if under a spell, it doesn't seem very real or emotionally significant, especially when compared against the supplication of Andromache and Hektor earlier which was so full of human emotion, and no real mention of gods until Hektor asks for the gods' blessing on his son.
All the trappings of supplication are there, the kissing of hands, "...Priam spoke to him in words of a suppliant." (Line 485). It all seems so formalized that it doesn't ring as true or meaningful.
Just my take....
All line numbers refer to Richmond Lattimore
's verse translation - University of Chicago PRess