If your interpretation ware to be correct, then every Delta Airlines pilot or Emirates Pilot flying in to the EU would need an EASA ATPL, but that is clearly not the intention of the Regulations, nor its effect. I recall a friend who flew for Cathay having to have HK ATPL because the aircraft were Hong Kong registered; likewise if you fly for BA, Air France Air India etc. That is logical, but I don't believe for one second that EASA intends that every pilot crossing its borders but working for a foreign carrier with foreign registered aircraft needs an EASA ATPL.
This was one of the objections raised very early to the EASA FCL proposals, and it would obviously show them to be a farce.
It was then suggested by various people that EASA would make an exemption for AOC holders, which would be easily done with a single sentence. This would have the effect of allowing AOC operators to carry on under ICAO provisions, while shafting non-AOC business jets and all the rest of GA below that level.
Yet, it appears that this proposal does not have such an exemption!