NOMTW, I think the point of misunderstanding between those pro and those against singing NA is the understanding for what NAs celebrate. You view singing the NA as recognition for the chayalim and America’s graciousness to the Jews. Those who oppose using the NA in Jewish events are of the opinion that an anthem celebrates the entire state, with all of its flaws, even those that the state itself doesn’t recognize as faults.
The reason we oppose using anthems is not because we are Kofeh Tov to the soldiers or to America’s kindnesses. It’s because the anthems celebrate the ENTIRE structures that are the USA and Israel, two countries (among many others) whose governments pride themselves (no pun intended) in allowing freedoms for homosexuality, providing funds, support, and land trade agreements to terrorist sympathizers, having a pop culture that is dedicated to the corruption of tzniyus, and provides a haven for those who speak out against Hashem and against the Torah.
If you sing the anthem you are praising an entire package, which includes things we are not allowed to condone. That’s why it’s far more appropriate to publicly state “thank you” or to say mi sheberachs which thank the governments in very specific terms, without condoning assur behavior and treif hashkafas.