You will earn lots of very useful BA tier points for your QR flights.

The issue is that all the FF programmes are differently constructed with different sweet spots. For example, BA gets very easy status for flying in Business (as long as you take 4 BA flights a year) whereas some of the US ones are less generous (they tend to penalise economy travel rather less). I'm familiar with the BA programme and aware that many Americans like it for its ease of earning status, the usefulness of that status in the US because of its partnership with AA, the ability to use lounges on domestic US flights and a redemption sweet spot for short haul domestic US flights. You sort of have to make the same analysis for all the FF programmes in order to determine which works best for you, but I wouldn't bother at all with those that aren't allied with one of the US carriers - since it would be difficult to benefit from that status.