First of all, there is no consensus that the sons of God who had mighty powerful children were angels, and not simply powerful evil men. Even if by some divine exception angels copulated with women, the Bible is very clear that the reason for the Flood was widespread evil, not necessarily DNA cleansing. Also, it is also clear in stating that all sentient beings with the exception of Noah's family died.
"And every living substance was destroyed which was upon the face of the ground, both man, and cattle, and the creeping things, and the fowl of the heaven; and they were destroyed from the earth: and Noah only remained [alive], and they that [were] with him in the ark. (Gen 7)"
It looks like there were some few giants in antiquity, but nobody knows if they were not simply genetic variations that appeared at random. Definitely, nowhere in the Bible it is stated that people like Goliath were descendants of Nephilim. Also, the Israelii spied who reported seeing a people of giants were basically cowards who exaggerated their account in order to discourage the invasion, and were severely punished for it. They were lying. A possibility is that the genetic, natural line of tall people was extinct, by disease or persecution.
However, the main problem is that there are several bad consequences with the Nephilim line of thought:
1) It creates two classes of people, one that supposedly can be redeemed and one that has "bad blood" or bad DNA. Jesus came to save the whole world. Every each individual. [Jesus] is the atoning sacrifice for our sins, and not only for ours but also for the sins of the whole world." (1 John 2:2 - also John 12:47, 1 Tim 1:15 and many other verses) Believing that only some can be saved is a sure way to prevent people from telling the gospel to them. Also, God blots out people from the book of life. Everybody is born with his name there, it is sin that removes it. (Rev 3:5) If you ever meet Lord Rothschild, be sure to tell him about Jesus` salvation.
2) It minimizes the work of demons and the power of individual choice. God gave us freedom of choice. We choose evil because we love sin, not because we have bad Nephilim DNA. (John 3:19, Eph 2:3). "This is the judgment: the light (Jesus) has come into the world, and people loved the darkness rather than the light because their works were evil. For everyone who does wicked things hates the light and does not come to the light, lest his works should be exposed." We are free to turn away from sin and accept Jesus' offer of forgiveness. Of course, some people have already made up their minds towards evil, and it's too late for them. Regardless, it was a decision.
Also, rebellion, witchcraft and sin will eventually lead to submission and control (possession) of people by demons. Those horrible crimes and perversions we see today are committed by people who are possessed. But they are only normal people who one day opened the doors of their souls to evil spirits.
3) It makes people waste time with foolish fables. Christians are supposed to focus on helping their neighbors, raising godly children and developing their spiritual lives. The Bible warns us about fables in several places: Have nothing to do with godless myths and old wives' tales; rather, train yourself to be godly. (1 Tim 4:7)
4) It robs God from His glory when we say His creation was flawed or that He has no power over Satan's capacity to taint it. Human beings were made in the image of God and their bodies were wonderfully made, including their genetic code (Psalm 139:14). All this talk about transhumanism, altered DNA does not come from God. I believe that possibly Satan will use the promise of DNA advancement to lure people to his government. Let's not help him in his advertising.
I've wasted some time with those fables, but not anymore. I believe Anthony's problem is that he doesn't believe in the accuracy of the Bible and because of that, he fails to understand the consistency and coherence of the whole. Jesus did and taught about it. You can't have Jesus and a flawed Bible. It is a whole package deal.
Try to find the core principle behind the doctrine: the world (everyone) has fallen because of love for sin. The world (everyone) can be saved by curing this disease through the forgiveness Jesus provided at the cross. No place for irredeemable DNA here!