Yes, we draw a flag for the stakeout point but it's not really a "flag", it's the symbol for the point. That's why when you turn off flags, the stakeout symbol (which looks like a flag) is still displayed.
The easiest solution is to just label the points with a label style that includes the "Symbol" type. In fact, the label style only needs that one type (the symbol) included in the label style. When the label style has a symbol defined, it replaces the default symbol for the point (the flag in the case of stakeout points).
This should be easier than copying the points.