The tobit and probit methods are quite different models; they should not be seen as slightly different variations of each other.
The probit method, as you have noted, only uses information about whether or not a country received aid. It uses no information whatever about the amount of aid.
The tobit method uses information about how much aid a country receives and incorporates the characteristics of countries that receive no aid.
The estimated parameters have no relationship, in principle, when most of the observations have an observed positive value of the dependent variable. Even when a small proportion of the observations have a positive value the estimates of the coefficients can be quite different.
The estimated probability that a country gets aid might have little relationship to how much aid a country gets.
The two models estimate quite different things, an amount of aid and the occurence or nonoccurence of any aid whatever. These might depend on different independent variables or depend on them in different ways.
It is clear that the two methods are designed to meet different objectives. Decide what your purpose is and you have the answer to what methods you need.
Oh, I forgot about your advisor's role in all this.
If you can honestly tell your advisor that your objective is strictly limited to estimating the probability that aid will occur and your advisor agrees that that limited objective is adequate for your research, then you can do only that.
You should pay more attention to your advisor. Your advisor is not only in a strong position of control over your success, which you might resent. Your advisor is also, presumably, knowledgeable about the field of research you are working in and is offering advice or instruction that will help you make a contribution to research.
If you are not comfortable with your advisor, you can look for another one. Ask around and see what other faculty think. Also, give careful consideration to the published literature and evaluate your own plan with respect to what other scholars have done.
Decide for yourself whether it is easier to disagree with your advisor, find another advisor, or just do two kinds of analysis, both of which are really pretty easy to do today.