Financial certification - for Int'l Students

I am looking at the financial requirements at some schools, and it is pretty confusing.

Most grad schools will give stipends, waive tuition fees, and provide assistantship opportunities. Without the kindness from the schools, most students are unlikely to survive grad school. At the same time, some of these grad schools require Int'l Students to certified their financial standing - I think it is also part of the visa application requirement.

Most school's require one to have enough funds to cover living expenses at least for the 1st year. I understand this. And if you are going with your spouse and child(ren), you need to add a few $k more. I understand this as well.

Now, what I don't understand, and that's main question, is how do we factor in the school's kindness (in the form of stipends, tuition waiver, etc.) in the financial certification forms?

If your personal (+ sponsored) funds BEFORE considering the school's kindness are insufficient to cover the living expenses, will it affect your ability to get admission or the visa?

Also a funny thing is that the minimum amount stated by the school usually includes the tuition fees, which is a substantial amount. But the school usually will waive these fees. Doesn't that artificially and unnecessarily jack up the requirement for the student?

Next, for visa, the application takes place after you have accepted a school's offer. By then, your financial situation will be different if the school extends her kindness to you (which normally happens). Do you then go for another financial certification when you are applying for the visa?

For those who have gone through this journey before, please share your experience. Especially if you were in one of the following situations when applying to the program:

1. You had enough funds to cover the minimum required amount LESS the tuition fees. Stipends and assistantships were a nice bonus.

2. You didn't have enough funds to even cover the minimum required amount LESS the tuition fees. But once stipends and assistantships (and tuition waiver) were granted to you, you were be able to meet the minimum amount required.