First, God created everything to reproduce after its own kind and angels are not the same as we are, in truth angles are made a little higher then us and without the ability to reproduce. But one day we will be made higher then them.
Not having a physical body with DNA would be a big problem for them.
Second, we have been given the right to become Sons of God as wrote in the first paragraphs of John’s gospel. Therefore men are Sons of God as well.
Plus if an angel breaks Gods law by creating evil men, it becomes a demon. (A word for bad angel’s)
And if that is not enough, I found the origin of this doctrine, the book of Enoch. This is a book that is full of that doctrine and replete with heresy to boot.
The book of Enoch exposed
Genesis 6 v4 (KJV)
There were giants in the earth in those days; and also after that, when the sons of God came in unto the daughters of men, and they bare children to them, the same became mighty men which were of old, men of renown.
God says men twice... does that mean 1/2 man demon beast thing. You know, it’s almost as if he were trying to say something.
shouldn’t the next verse read
Genesis 6 v5-6 (NOT)
And God saw that the wickedness of 1/2 man demon beast thing was great in the earth, and that every imagination of the thoughts of his heart was only evil continually. And it repented the Lord that the angels had made 1/2 man demon beast thing on the earth, and it grieved him at his heart.
I believe that the most likely scenario would be that Godly men in right relationships with God lived so long that they failed, as we all do. Looking upon beautiful young women they became consumed with lust and broke the yet unwritten commandment of adultery.
One day we may know for sure what really happened but for now this answer is fair enough.
Oh yea... there were giants in the world in those days but it doesn’t say there were short people. And it took 150 years after the flood (in Peleg’s day, when the world was divided) to cut the life spans down to about 250 years, That’s why the bible says that the giants were there also after the flood, because the family on the ark were giants as well: with the new atmosphere after the flood, I’ll bet that some strains of DNA retained the size of people better than others and that’s why there were giants all the way up to David's day.