As I understand it, in order to have real end-to-end encryption (the real stuff, not some marketing term) each device has to generate a long set of keys and with each message sent, they cycle through to the next key. If WhatsApp is doing what it reports it's doing and it actually is end-to-end encrypted then the web application needs to use your phone because it needs that set of keys. I'm not sure if it specifically sends through your phone or if it sends via the webapp, but you have to use the keys in the correct order or the device you're contacting won't be able to decrypt the next message.