The sentence conveys that better opportunities are available there, not that the country makes an offer. This makes offer inappropriate. Provide is similar to supply, which is different in context to give, avail and offer. The US is supplying/providing opportunities. It is not offering, but offering is similar to ask. It is not asking foreign students to take the opportunitiies. Provides is more direct than gives and more specific. Avails is incorrect because it is students who can avail the opportunities, which the US provides.