This is a pretty basic question.
Since browsers have a culture setting that the web app uses to decide which locale to use, should I still have a "language" button for users to be able to override the culture?
Why or Why Not?
To me it doesn't make sense to have a button there if the user had already set their language in their system.
Yes, you should.
One of the reasons outlined on the ASP.NET Localisation page is that a user may have a preference for reading certain types of websites in a specific language (i.e. technical websites in English). This is a good reason.
There are other reasons as well (i.e. the translation may not be great, a different user than typical is using the given computer, etc, etc).