This seems to be the default belief of the early church. At least four of the church fathers spoke about this and agreed with it, including Justin Martyr, Athenagoras, Lactantius, and Commodianus.
It's pretty clear that Jude is referring to some of the events described in Enoch in his letter. That isn't to imply that the book is equal to scripture, or even that Jude agreed with all of its contents. But Jude, and also Peter, seem to be referencing that tradition.
Afaik, it's less clear to me that demons are the spirits of dead Nephilim. But that belief was widely held in the early church. And I don't think demons are ever presented as corporeal, as angels are, so I'm inclined to believe that they're different classes of beings.
When Jesus rose from the dead he said that "All power in heaven and on earth has been given to me". That is, the angelic principalities no longer have any dominion over men; God has reclaimed for himself all of us, every nation. We are truly living in an age of grace.
Dominions Principalities Powers still exist. these are not human beings or human institutions.
Colossians 1:15-17 New King James Version (NKJV)
15 He is the image of the invisible God, the firstborn over all creation. 16 For by Him all things were created that are in heaven and that are on earth, visible and invisible, whether thrones or dominions or principalities or powers. All things were created through Him and for Him. 17 And He is before all things, and in Him all things consist.
Paul wrote Ephesians after the resurrection, and describes our fight against evil. Notice the devilish connections called the rulers of the darkness of this age. That age we still live in and the fight goes on.
10 Finally, my brethren, be strong in the Lord and in the power of His might.
11
Put on the whole armor of God, that you may be able to stand against the wiles of the devil.
12 For we do not wrestle against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this age, against spiritual hosts of wickedness in the heavenly places.
13 Therefore take up the whole armor of God, that you may be able to withstand in the evil day, and having done all, to stand.
14 Stand therefore, having girded your waist with truth, having put on the breastplate of righteousness, 15 and having shod your feet with the preparation of the gospel of peace; 16 above all, taking the shield of faith with which you will be able to quench
all the fiery darts of the wicked one. 17 And take the helmet of salvation, and the sword of the Spirit, which is the word of God; 18 praying always with all prayer and supplication in the Spirit, being watchful to this end with all perseverance and supplication for all the saints— 19 and for me, that utterance may be given to me, that I may open my mouth boldly to make known the mystery of the gospel, 20 for which I am an ambassador in chains; that in it I may speak boldly, as I ought to speak.