How did you setup the application pools and/or ftp?
As by default the _web user indeed does not have write (to protect the site as it’s used by IIS to run the web files)
The _web user however is normally not the same as you would use for Web Deployment or FTP, neither is it used by the Web Based file manager.
In all those occasions you by default assign read/write to the users you use to upload.
If you use a website it self to upload/change then you would need to set write permissions to the wwwroot folder with inheritance to the _web user (or in policies set writeable)
Do note that that will have impact on the security of your website.