Whatsapp don't retain messages/media after they've been delivered to your phone, which is a compelling privacy feature for many.
It's also what allows them to serve such an enormous user base with limited hardware. Their technology stack (FreeBSD/Erlang) is pretty interesting, more info here:
2014 talks by Rick Reed:
https://www.youtube.com/watch?v=c12cYAUTXXs
https://www.youtube.com/watch?v=TneLO5TdW_M
Slides:
http://www.erlang-factory.com/static/upload/media/1394350183...
There's likely no technical reason why you couldn't use a pc instead of a phone for users that want to use the pc as the primary client (with the phone optionally accessing the DB on the pc in the same way that the desktop client does for the phone). Perhaps they've decided that this is a small and declining market.
Edit:
Slides for second talk
http://www.slideshare.net/iXsystems/rick-reed-600-m-unsuspec...
440k connections/sec, 1.1 million msgs/sec, 1 billion images/day, and that was in 2014...