Not only that. Post-Colombian food _is_ an integral part of many countries’ cuisine now.
Italian food without tomatoes is nigh unthinkable. And potatoes form the base for so, so many countries.
There are other influences too. The aubergine was first cultivated in South and East Asia but today it’s a staple in the Mediterranean and Middle East as well.
At some point one has to stop thinking of indigenous food sources as the only way to form a countries’ cuisine.
It is also interesting to realise just how little time has to pass in order to make a practice inseparable from a people. Some aspects of Italian coffee culture are quite recent, for example.