That doesn't work, as anyone providing you a clientside cryptosystem can provide you a backdoored clientside cryptosystem at the government's demand (one that silently uploads your key material to the server).

It doesn't matter if they don't normally store the key. It's a webapp.

Also, they need the key to do search. Furthermore, this does nothing to hide the metadata surrounding your communications, which necessarily must not be encrypted for services to work.