I'm not sure what the hesitation to get a gift is...
I think I understand from the American point of view. My group is predominantly Chinese international student. I am Canadian of east Asian descent and I do understand Chinese, for the most part...
At first, I tried to organize events and outings for the international students, trying to help them feel at home. After a while, I realized that there is so much negativity around the students, constant complaining on how convenient China is, how things don't work as they know it, how much they miss their home, and yet they all come here with the goal to obtain a citizenship. Can you imagine how insulting and irritating it is to insult another person's home in their home? On top of that, just the general negativity of conversations makes me uneasy. I do feel embarrassed to take them to restaurants and places because of the cultural differences and language barriers.
My experience is that international students of same nationality/language will always clump together with little to no motivation, perhaps due to external resistances, to move out of their circles.
At the end, while I feel that it was nice for me to help them but it is not my responsibility to help them integrate.
Anyways, just sharing some of my thoughts and experiences and also to rant.