I certainly agree that context matters a great deal. As Greg Koukl says, context is king. The irony here is that the author has inserted his own context into Romans 13. There are several concerns but none of them are about "Judean nonbelievers’ and Gentile believers’ relationship". His concern for unsaved Jews was for their salvation, not their relationship to Gentile believers. The main theme of Romans is the gospel, as seen in the first few verses, for both salvation and its practical outworking in the Christian life. After giving the theology in the first 11 chapters, he then provides the practical application in the final ones. The Jewish and Gentile relationship that Paul is concerned about is of that between believers (chapter 14).
The church in Rome was likely mostly Gentile but also had Jewish believers. There is nothing at all in the context of the entire book of Romans to suggest that 13:1-7 is speaking of the rulers of the synagogues, and everything to suggest that he is indeed speaking of the Roman authorities. This is made all the more clear by the commands for general Christian conduct in chapter 12, which ends with not taking vengeance, doing good to one's enemy, and overcoming evil with good. To believe that Paul then suddenly switches to a discussion about obeying synagogue rulers does not make sense. Paul is continuing the same line of thought from the end of chapter 12, where it is the Roman authorities (and, by extension, all authorities, everywhere, in all times) who are God's chosen instrument to punish the evil doer. Chapter 14 then begins with discussion about passing judgement on what other believers eat and drink and what days they observe. This strongly suggests disagreements between Jewish and Gentile believers. Again, nothing to do with synagogues or Jewish rulers.
One glaring error by the author is this argument: “This Judean context makes perfect sense of verses 1-7 as one reads Romans. For example, Paul speaks of authority that exists from God (v.1) and is appointed by God (v. 2). This hardly sounds like a description of Caesar and his predatory legions.” Yet, Jesus says this to Pilate, a Roman governor:
Joh 19:10 So Pilate said to him, “You will not speak to me? Do you not know that I have authority to release you and authority to crucify you?”
Joh 19:11 Jesus answered him, “You would have no authority over me at all unless it had been given you from above. Therefore he who delivered me over to you has the greater sin.” (ESV)
See also Hab 1:6, which shows that God raises or puts in place even evil governments and rulers for his purposes, and Dan 2:20-21, where it is God who both removes and puts in place kings. That the author would make such a glaring mistake suggests that he is reading his own ideas into Rom 13:1-7.
That Paul is speaking of earthly authorities in Rom 13 is supported by his instruction to Titus, to “remind [the believers in Crete] to be submissive to rulers and authorities” (3:1). That is entirely consistent with his command to the church in Rome.
Additionally, we also need to consider what Peter wrote:
1Pe 2:12 Keep your conduct among the Gentiles honorable, so that when they speak against you as evildoers, they may see your good deeds and glorify God on the day of visitation.
1Pe 2:13 Be subject for the Lord's sake to every human institution, whether it be to the emperor as supreme,
1Pe 2:14 or to governors as sent by him to punish those who do evil and to praise those who do good.
1Pe 2:15 For this is the will of God, that by doing good you should put to silence the ignorance of foolish people.
1Pe 2:16 Live as people who are free, not using your freedom as a cover-up for evil, but living as servants of God.
1Pe 2:17 Honor everyone. Love the brotherhood. Fear God. Honor the emperor. (ESV)
There is absolutely no doubt that Peter is here telling believers to obey all earthly authorities. But what is even more significant is that this was during a time of persecution, which was rather one of the main reasons for the letter. This, too, is completely consistent with Paul’s commands to the church in Rome to obey the secular authorities.
What does all of this add up to? It means that all believers are to obey all earthly authorities as God-given, with the obvious exception that we do not do so if what they require would cause believers to sin.
There is more that can be said but, for the sake of brevity, that is sufficient enough to show that the author of that article is making up his own context. The stronger position is that Paul really is telling the believers in Rome to obey the Roman authorities because God has placed them there. That is a consistent message throughout the Bible.