What is the function of the payload_checksum field in the Bitcoin protocol
"Full protection" is a relative term.  TCP packets include only a simple checksum, effectively just the sum of all the 16-bit words in the packet.  See RFC 793 section 3.1.  This is a fairly weak integrity protection mechanism.  The Bitcoin designers may have felt the use of sha256 gave better protection.  It may also, as you say, have been intended to provide forward compatibility for running Bitcoin over UDP or other protocols.  It may even just have been paranoia.