When we say that we are saved by faith (belief, trust) and not by the law, we simply mean that we are saved by believing in Jesus or by trusting Him and no longer to the works of the Law of Moses. That is John 3:16.
We are not saved by the law implies that Jesus already fulfilled that law (in the shedding of His divine blood) which cannot save us. That law is nothing but the works of the law, as it is stated in Galatians 2:16.
The works of the law by which we are not saved include the tandem of the slaughtering the animals to sprinkle their blood, the rites involved in the slaughtering, and the related sabbaths brought about for the rites. Because Jesus had been slaughtered, that part of the works of the law can no longer be observed. That is the message of the Apostles in Galatians 2:16 to the Jews to teach them that Jesus is the works of the law and that they must believe (have faith, trust) in Jesus the Messiah to get full justification rather than the daily or annual justification that was provided by the works of the law of Moses.
Justification by faith (belief, trust) is easier to us Gentiles or Pagans than it is to the Jews because they are to be convinced to give up the works portion of Moses and take Jesus as the new mean of justification to be saved. Remember the statement of immutability that all of us must hold firmly to the letter according to Deuteronomy 4:2, 5:32, 12:32. The Baptism of John the Baptist was to soften the stance of the heart of the unbelieving Jews to understand the temporal nature of the works of the law of Moses.
JESUS DOES NOT CHANGE MOSES AND THE PROPHETS IN ANY WAY BECAUSE JESUS SAYS HE DID NOT COME TO LOOSEN THE LAW AND THE PROPHETS in Matthew 5:17-18. How can men teach a loosed Moses in matters of faith, food, and other things which I am not aware of?
Compare the problem posed by Isaiah 65:4 and 66:17 which rebukes and condemns the eating of forbidden pork. Jesus was speaking about the washing of the hands before consumption of food that does not defile. Jesus was not talking about the consumption of unclean food (Matthew 5:10:20, Mark 7:1-13).
Matthew 5:17-18 ASV
(17) Think not that I came to destroy the law or the prophets: I came not to destroy [loose], but to fulfil.(18) For verily I say unto you, Till heaven and earth pass away, one jot or one tittle shall in no wise pass away from the law, till all things be accomplished.
Matthew 15:16-20 ASV
(16) And he said, Are ye also even yet without understanding?(17) Perceive ye not, that whatsoever goeth into the mouth passeth into the belly, and is cast out into the draught?(18) But the things which proceed out of the mouth come forth out of the heart; and they defile the man.(19) For out of the heart come forth evil thoughts, murders, adulteries, fornications, thefts, false witness, railings:(20) these are the things which defile the man; but to eat with unwashen hands defileth not the man.
The Jews in the time of Jesus and of the Apostles kept Moses. The Gentiles also kept Moses as we can learn from Acts 15:21.
Today and tomorrow, not all believers keep Moses as that was the case for the first Christians of the Apostles. Today, we pick and choose what we like from Moses to stand out from Judaism. We do not even confess who Jesus is to not die in our sins (John 8:24).