The B767 cross feed did not meet the reliability required hence the original requirement to open the cross feed at the EDTO entry point. This was deleted once the second cross feed valve was available and installed in the aircraft.
The B737 had two APU suppliers one of which was able to demonstrate air start reliability, the other required the APU to be on at the EDTO entry point.
EDTO and MNPS are two different fish. Loss of a LR Nav system does not affect EDTO.