I'm not in a position to say that no better way is possible, but all ways of defining economic aggregates run up against conceptual problems when you try to apply them to an actual economy, the most intractable of which is how to assign a single number to a changing mix of heterogeneous goods. The difficulties with defining "inflation" when the basket of consumer goods changes every period is well known, but you have similar difficulties with defining production if you change your inputs -- of which outsourcing is a special case.

In most situations you end up using price as an aggregator, even if the price is itself normalized by a scaling factor. That means if you shut down a factory today and have your goods produced cheaply overseas except for a few small finishing steps here, it will be the relative prices of the inputs that determine how much of the good was produced here versus how much was produced overseas, and this wont generally reflect the "real" amount produced here versus overseas as viewed by a layman. In fact, if a producer couldn't outsource cheaply, they probably wouldn't do it, so there will be a consistent gap in terms of what the laymen sees and what the BEA data will show. And note this also creates the appearance of a productivity boost. Imagine Apple firing all their manufacturing employees and employing only a few designers. Now, they can produce a lot of value with only a few cost inputs, hence productivity just went up. Of course, the designers may not be more productive in terms of creating more designs or better designs, but as Apple is able to capture a large share of the total value, then it will be attributed to the super productivity of the designers, which may have more to do with politics in East Asia choosing an export led growth model than anything to do with the iPhone.

And it gets more complicated because the NIPA system doesn't take into account the possibility of economic rents. E.g. going back to the Apple example, if Apple can charge $1000 for the phone, with only $300 of inputs, then this $700 worth of "value added" must be attributable to some portion of labor and capital. If the designers are paid $200 then capital is paid $500. In NIPA parlance, that means $500 worth of capital investment (and or depreciation) must have occurred somewhere in the economy, which again to the layman would require unrealistic measurements of investment. There is no NIPA notion of someone overpaying. And then don't get me started on how corporate shell companies headquartered in tax shelters like the Bahamas or Ireland screw with GDP numbers, especially of the smaller nations that serve as tax shelters, but also with US productivity and investment data.