I’ve honestly never given the matter of a flag in a church any thought.
Our church doesn’t have one, but for the most part, you won’t find a more patriotic bunch. We have at least two flags in the building. AWANA (kids group) uses one and trail Life USA (Christian boys outdoor adventure group) uses one. We also have several more that we use for flag folding and when camping.
Patriotism isn’t encouraged or discouraged, it just isn’t really part of the church setting.
We’ve always had the American Flag on the left side, and the Christian Flag on the right.