Not an excuse, but: A) WhatsApp is a true messaging app. Once the messages are delivered, they are deleted from the server, so you need a primary device that stores the messages. B) It currently uses a phone number as identification, in order to create a true desktop app you would need another method (unless you use a landline or dumbphone phone number for desktop as well).
Messages are not deleted from the server. They are probably stored there in the encryped form. When I open the web version of WhatsApp, same images and the message history that i have on my phone get downloaded there. I highly doubt that they rely on my phone as the only master storage of that data and upload all images into the web version from it?
My experience of WhatsApp Web has been that the images do not auto-load (Chrome on Windows 10 and Windows 7). A heavily compressed thumbnail is loaded and I have to manually click each image that I want to view.
I haven't analysed any traffic and I'm not a developer but my layman's guess would be that the WhatsApp mobile app has a pre-prepared Zip file (or some other compressed container) with the 20 most recent messages from each chat plus these compressed image thumbnails and the Web Client pulls that data from the phone upon initialising. I doubt it is a big data transfer.
Using mitmproxy to snoop my phone, I see that all the data coming into my computer when I open up Whatsapp Web is originating from my phone. Including pictures.
That being said, all of my current conversations are on the new "end-to-end-encryption" thing, which might make a difference.
Images and videos are stored on their server for a fixed amount of time, and deleted afterwards. (So you just send an url and the client transforms it into a thumbnail).
You can test this by trying to download an image for the first time after a few weeks, it will tell you that the image is no longer available, and you should ask the other recipient to send it again.
That's not true: I just looked in my WhatsApp gallery, the first image I opened was 525KB. At <100KB, any screenshot of text would show visible compression distortion around the letter edges, which they do not.
I think you should try web app disabling the internet on your phone. Have you tried it? If watsup store the messages on its servers, it would be able to retrieve. But this is also not reliable since they may check phone connectivity before they retrieve from their "servers". I am still skeptical that they may store on servers behind the scene at least for speed and performance.
I just opened the Whatsapp web app (never used it before), and then immediately set Airplane Mode on my phone as soon as it scanned the QR code.
WhatsApp managed to load all my conversations (I don't have many) and the most recent message, for display in the list-view. But each conversation only has the most recently sent message. After a while I got a "Phone Not Connected" message. Seems like it really is only stored on your phone.
In order to use Whatsapp Web you need to be on the same wifi network as your phone.
Your phone has the data on it. Intranet speeds on most wifi networks these days is faster than 50Mbps. That's more than fast enough to transfer all your messages and photos with little to no lag.
It doesn't matter how fast your connection to the WWW is because you're not using the WWW to transfer the data, you're using the wifi intranet.