Yes we do fly safe- you will not find a safer airline or one with better training, believe me....and certainly not one in the world with more experience of worldwide longhaul heavy jet flying than BA. We actually know how to do it pretty well because nobody has done more since 1945.
If it was LHR-LAX when an engine failed, it would most probably have returned to LHR before PNR as that would be easiest for repair. After losing an engine over Caraffa, I returned to LHR- it is actually easier to do that and transfer a to another aeroplane than dump them at Rome or Athens at 2 am to hoe to get a flight on later that evening! But this is hypothetical- we are talking about LAX-LHR.