Many issues with these opening statements, but first maybe, is that since a common σ calculation is but a special case of a GARCH set-up (for example, the limit as measured heteroskedasticity tends to zero), and any GARCH-calculated σ is going to carry biases by construction. Why would you expect otherwise?
Second, realized vol (whether from a common σ or as some GARCH byproduct) is an outcome computed from history. It is not an estimate; it is realized. If you wish to draw differences between the GARCH product and the common σ, have at it[!!] But unless you are specifying a population of data, and declaring a subset as the relevant lens, you have a population parameter, not a sample statistic.
Third (and of related) concern: "When do we hedge to realize GARCH vol?" is not a statistically valid statement. It is the market which moves, and the market through that movement which defines volatility. We only realize mistakes. (If we're lucky.)
Best wishes.