"Cherry pick from religious scripture" is religious thought(s). It's approaching politics through religion. Christian communism is the same thing. However, saying political ideas are mistaken because they're interpreted through religion is more of a question mark, because it could easily be considered politics and not religion, depending on where one draws the line. Unless debating the idea of religion is considered religious thought, unless doing it through the lens of politics/political theory/political philosophy makes it political.
If this helps (I doubt it will), but Christianity is neither right-wing or left-wing. IMO, the political talking points of the current day are only shadows of what Christ has in mind. And this would be expected. It would be expected that humanity as a whole doesn't get things totally right according to God's standards for politics. In short, there are good things about the right-wing and good things about the left-wing. But God is neither exclusively right-wing or left-wing. God is very merciful to those marginalized, but the Bible also constantly calls for order. In this way, it is a balance, and it is an error to fall too far on either side.
There are many ways I could show this, but one clear way is from this passage of the Bible I will quote here...
Matthew 22:15-22
"Then the Pharisees went and plotted how to trap him by what he said. So they sent their disciples to him, along with the Herodians. “Teacher,” they said, “we know that you are truthful and teach truthfully the way of God. You don’t care what anyone thinks nor do you show partiality. Tell us, then, what you think. Is it lawful to pay taxes to Caesar or not?”
Perceiving their malicious intent, Jesus said, “Why are you testing me, hypocrites? Show me the coin used for the tax.” They brought him a denarius. “Whose image and inscription is this?” he asked them.
“Caesar’s,” they said to him.
Then he said to them, “Give, then, to Caesar the things that are Caesar’s, and to God the things that are God’s.” When they heard this, they were amazed. So they left him and went away."
The point is that you had various religio-political forces at work trying to force Jesus to take a "side." The Pharisees were opposed to Caesar. The Herodians were loyal to Caesar. These are a couple of the groups that were at work in this discussion. Jesus doesn't say, "I follow Caesar," or "I am against Caesar." He says, "Give to Caesar what is Caesar's and give to God what is God's." The natural implication is that God appointed Caesar to be in power at that time (John 19:11), but God USES those in political power to do His will. Ultimately, God is in Charge, and we need to remember that Christ will ultimately be in the judgment seat and will judge the nations.
Proverbs 21:1
"A king’s heart is like channeled water in the LORD’s hand:
He directs it wherever he chooses."