Seen in the light of the end of Romans 12, this is for Christians to not do violence or violently resist authority, knowing that those who are there are allowed to be there. It in no way forbids democratic republican systems of government. God put those there (or rather: allowed them to be) also.
This section of the chapter is talking about living a holy life in the empire of Rome, we no longer have that empire, but the principles are the same. With me, an American, "honor the president", submit to the authority, etc. We also know by the fact of the common condemnation of the innocent, that it isn't really the only purpose of the government to "punish evil-doers," and the fact that Jesus was condemned, and martyrs made, we can know that they do not exclusively "punish evil-doers." Either way, Peter here (not even Peter, but still) is making this the main point: "Live as free people, but do not use your freedom as a cover-up for evil; live as God’s slaves." which goes into the next points of Scripture you didn't quote, but I will since your reference to 1 Peter 2:18 is essentially the same:
The famous "slaves obey your masters, wives obey your husbands, children obey your parents," etc, etc. The purpose of this chapter isn't what you seem to think it is, but is a reminder of Christian citizenship. We aren't members of this world, so St. Paul says: "If then you have been raised with Christ, seek the things that are above, where Christ is, seated at the right hand of God. Set your minds on things that are above, not on things that are on earth." Therefore our lowly stations in life (of which were the majority of Christians at the time, the poor, the slaves, women, children of pagans, etc) do not matter, so we shouldn't do evil to those above us, but should show Christians submission to their authority (although clearly not obedience if it contradicts the law of God, which it often does, hence martyrs. But even in disobedience we show submission, accepting the punishment). That is why he ends with: "Whatever your task, work heartily, as serving the Lord and not men, knowing that from the Lord you will receive the inheritance as your reward; you are serving the Lord Christ. For the wrongdoer will be paid back for the wrong he has done, and there is no partiality." Those above us who wrong us with violence and oppression will be paid back, but because there is no partiallity, if we wrong them, so will we. Therefore, we don't pay back evil for evil, but good for evil (Romans 12).
Someone being elected doesn't mean that the power isn't from God, this is clearly not a comment on political systems, but power over Christ (and Christians) in general. It is given by God, so if it happens by lineage, so be it, if it happens because people put them there by consent, so be it. God is the Lord of history, if He really had a bone to pick with democracy (as in it went contrary to His sovereign will for history), He wouldn't allow it.
Barabbas is spared thanks to the sin of the people and the will of God. Check the commentaries on this passage from all throughout the ages, it has nothing to do with democracy. People can choose evil in/under any system. You could say Barabbas is spared thanks to Pilate even giving them this option, and accepting it. He had the decision, he chose evil. Is it thanks to the empire now?
This literally doesn't say this, but ok. Also it was an appeal they accepted, is bringing a complaint to the king not good in your idea of a monarchy?
This is absolutely true, and agrees with St. Paul. I have no real comment on this interpretation of reality tbh, though I do not think this implies what you bring it to here: "embracing slavery" under a king.
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