This isn't part of my Ten Commandments series (not yet), but it is related. BTW, Boxty recommended Prager University's series on the Ten Commandments and it's quite good. It is very basic, but I learned at least one new thing from each so far. It is clear that Prager knows much more than he shares.
Nationalism, the flag, or now, the anthem, can be a false god, certainly. We see that clearly when it's some other guy's nationalism.
Conservative Christians often hesitate to go there and warn about it for a few reasons. First, most warnings against this particular idolatry are insincere. They come from liberal Christians who have their own idol they would prefer us to worship, internationalism. I don't know how clearly they see that there is danger and possible idolatry in this belief, far more than in nationalism, but my observation is that they do not see this at all. Internationalism is about co-operation, and respecting other people, and being open-minded, and all those other nice things that Jesus clearly would rather have us do than dumb old nationalism, with its Nazis, and unfashionable flags. If they do get it, I never hear them mention it or warn each other about it. I have to conclude that in their hearts they really believe that internationalism is much closer to God's plan for mankind. They speak and act as if it is holier, more Christian, than patriotism of most sorts. I would assert that it is far more spiritually dangerous.* I think many conservative Christians would agree, and thus dig in against anything that would bring us closer to that heresy, to the point of tolerating a little idolatry of our own.
If it is not any more dangerous, it is at least no better. It is like those advisors who tell us it would be better for our health if we ate less beef. Perhaps so, but I see behind you a fishmonger's cart. Thus, I suspect you are telling me this for your own advantage, not mine.
Next, American nationalism (some other nations deserve mention here as well) considers itself something of an ally of the Christian church and has learned to regard Judaism as a co-ally as well. Many folks tie these two together quite tightly. This was even more true in earlier times, though the alliance was more specifically Protestant Christian. This has broadened (mostly), and as a consequence become less automatic. Still, it remains powerful. Those who are the most loyal supporters of one are often the most loyal supporters of the other. Even Christians with a touch of fanaticism hesitate to kick their friends unless they think they really deserve it.
Relatedly, American freedoms are a protection for the church, and it would seem ungrateful to bite the hand that feeds us. It is true that it is a great blessing to be able to worship without fear, and people fought and died that we might have that gift. Yet it would be more accurate to say that American religious freedom is the current method by which God protects his people. He has accomplished this in other ways at other times. As the freedom may be rescinded without warning, we should not think God is unable to adapt to some changing political circumstance of ours.
Christian groups which have separated themselves most thoroughly from Americanism or other nationalisms - Armstrongites, Jehovah's Witnesses, even some of the peace churches - have been held to be somewhat outside the faith. (Though note, the pacifism of the Early Church was often of this kind, regarding no nation or government as worthy of that degree of loyalty, as contrasted with an opposition to the use of the sword in all circumstances.)
Lastly, I think that conservative Christians often do not see the conflict, just as liberal internationalists don't see any contradiction. We are not the only nation to have an explicit phrase "For God and Country" - it is even the title of the Scout award for studying your religion - but being more powerful and having religions that are shared in other countries, at least in name, it gets a lot of attention. I know people for whom both pieces, plus some cultural things from their region or ethnic group, is regarded as being traditional or conservative, without clear boundaries as to what is God's, what is Country's, and what is Culture's. They are more muddled than rebellious, yet it is ripe ground for idolatry.
With all that said, I believe the danger of patriotism or nationalism being an idol is real, and much more so for conservatives. As a young Christian I looked askance at the notion of "pledging allegiance" to anything but God, and still think the common understanding of that goes too far down the road to muddled loyalty. Pledge is a less strong word than vow, more strong than loyalty; it still carries a lot of weight. One rescinds a pledge only in extreme circumstances. Allegiance does not come from "ally" but from "liege." (They are related farther back.) A word's origin does not dictate its meaning now, but it often explains a shading. Pledging allegiance is a declaration of loyalty and subordination, that we will put the country's need above our own. Such intermediate loyalties - to family, to tribe, to master - have always been allowed in the Christian church, but they have been long regarded as possible competitors. Jesus gives his mother and favorite disciple to each other as family, honoring that loyalty; Paul declares that people should support their families. Yet Jesus also declares that when pushed to the end, one must leave the dead to bury the dead; the new tribe that is the Church are the real mothers and brothers. Other loyalties are allowed, but they are limited.
There may come a day when America is notAmerica, and the flag is used as a manipulation to bind those who have been raised to revere it to an entity that no longer deserves it. There are always those who say that time has already come. We may be assured that this will eventually come to pass, as all human things do eventually fall away. Pray that those who live in those times have wisdom and courage.
*I should probably post on why that is rather than merely assert it. For the moment, the fact that it is harder for those believers to even see the distinctions is part of the answer.