How does already having an academic scholarship affect graduate school admissions?
Having your own funding may help, e.g. if you are a good match for a faculty member who has an interest in supervising a PhD student, but does not have enough funding to support one at the moment. But no department wants to waste time and resources (e.g. supervision resources, for example) on a student that is not capable of meeting standards in the department. And given that a department's PhD students represent its "brand," no department wants to admit a student whose work will reflect negatively on the department.

So yes, even if you have your own funding, you still need to show the department you're applying to that you're capable of meeting their standards.