Don't know about the 777F however -400F and -8F certainly have a W & B computer onboard and, yes, we rely on it to the extent that if it is 'out' compared to the load sheet there are procedures to follow to ensure an error hasn't been made.
MEL allows you to go without it but beware! In my experience, (30 plus years although some of the early versions on other types were not the most reliable) every time it has been in error on the 400 or -8 there usually was an error! Many F/Os I fly with comment to me "Oh, but that error is common, CN so and so the other day just dismissed it and we flew OK". That is only a testament to the certification. I've seen many a Ground Engineer try and sign it off as unserviceable for no other reason than it was telling him something he didn't want to know. Subsequent checking of the load usually proves that the computer was telling the truth and there was a problem with the loading.