Very simply for the purpose of the OP question, what is the blood? What does it do?
"For the life of the flesh is in the blood: and I have given it to you upon the altar to make an atonement for your souls: for it is the blood that maketh an atonement for the soul." Lev 17:11
Is not the life of the flesh according to modern medicine still effectively in the blood? If all of a person's blood is lost is not natural life also lost?
So what was Jesus talking about here?
"...Verily, verily, I say unto you, Except ye eat the flesh of the Son of man, and drink his blood, ye have no life in you." John 6:53
I would say that the life as per the Book of Leviticus was NOT the same as the Life as per the Book of John. The former is only shadow using the body of man's flesh as a type of that which was to come [the old man versus the new or inner man]. The latter is the real thing, the Life which was lost to mankind when Adam and Eve disobeyed God and lost access to the Tree of Life.
We need to do more than "plead the blood". We need to drink [consume] the Blood, and this is likely not speaking of the red blood that flowed through the body of flesh of the man Jesus who walked planet Earth for about 33½ years some 2000 years ago, is it? The Body is that which needs to have Life is the Body of Christ, which is what we need to want to be part of...
Consider what the Apostle Paul was saying with these written words:
"Who [God] also hath made us able ministers of the new testament; not of the letter, but of the spirit: for the letter killeth, but the spirit giveth life." II Cor 2:6
It seems to me that the Blood we are to drink is the Holy Spirit for it is the Spirit that quickens and brings to Life within us the raw flesh of Jesus which we consumed when we read or heard the scriptures. Remember another verse already quoted in this thread:
"So then faith cometh by hearing, and hearing by the word of God." Rom 10:17
Where does a man get the "eyes to see" or the "ears to hear" but through the Holy Spirit?