Since your scheme is separate from Bitcoin, it seems that it would be cumbersome for users: they would need to run a special client
Well, it might be sort of a plugin for one of clients...
There are different clients, though, and many people use online wallets. Just something to keep in mind (eg you would not want the online wallet to know which addresses are yours and therefore link them!) That said, an easy-to-use plugin or standalone application is probably the way to go, sure.
Not necessary. If a lot of people use mixing, one round can be done in less than a minute.
not sure if that fast, don't forget that the batches will be queued up and broadcast in some random order. And it will take time before "a lot of people" are using the service on a regular basis!
Also, how is the resulting mixing more secure than the server simply selling people Chaumian cash?
Isn't it obvious? You can keep your bitcoins securely in your wallet. Service simply helps to negotiate a transaction which sends Bitcoins from your wallet to your wallet. There is no risk whatsoever.
I meant secure as in indistinguishability of outputs. You are right, of course.
"Simply selling people Chaumian cash" would require people to trust that server, develop a way to pay with them etc.
In which sense it is simpler than running a "mix my coins" plugin for your wallet?
Not simpler, but having some of the mixing bypass the Bitcoin network completely is potentially more secure.