The problem is that those precise measurements don’t really tell you much about the actual quantum state you’re looking at. Like the chord example, a position state is made up of a range of many momentum states. Unintuitively, the fewer position states something is in (the more specific the position) the more momentum states it is also in. But what you’d really like to see is a precise position measurement on A and a precise momentum measurement on B. The Uncertainty Principle doesn’t say “you can’t simultaneously measure position and momentum with nigh perfect precision!”, it says “it doesn’t matter if you do!”.