Azure App Service is a service used to create and deploy scalable, mission-critical web apps.
I could use a little bit of clarification @Борис Мкртчян . I'm assuming your updating static file content (e.g. html, css, js, etc.), so when you say...
We assume that the publishing profile on the admin site somehow leads to the "wrong" "wwwroot" folder
... you're referring to getting deployment credentials. I want to make it clear this publishing profile doesn't have anything to do with updating the static files in wwwroot. Furhtermore, there is no wrong wwwroot folder, as there's only one per the App Service file structure.
You can update your static files using the Kudu console. Try restarting your app if you're not seeing your changes immediately. However, I would strongly encourage using a continuous deployment method even if your is only comprised of static content.