The whole situation is fairly incestuous (a bit like the old money rich). If an English ‘french’ cook improved something, it would get rolled into ‘french’ cooking. It could even flow back to the ‘french’ cooks in France, given time. At this point we just don’t know anymore how information flowed.

On top of that, anything that seems to match the French style gets thrown in with French cuisine. Whether it was actually a French invention or not doesn’t matter.