The OP's question appears to be how "loyal" would anyone be to BA without BAEC, not whether they would fly at all. The Greenpeace quote in the BBC article seems to be more specifically around Avios/miles in general, not sure whether the other benefits of FFPs are in question.

It wouldn't stop me flying BA, but like any other purchasing decision, if a benefit is removed it could be a factor when buying a given flight, especially if all other things are more or less equal. The importance of BAEC to BA is well-documented around these parts, so one would imagine it would affect business to some extent, if BA relies on its FFP for business more than other airlines.