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...
and keep it online for a long time.
Not necessary. If a lot of people use mixing, one round can be done in less than a minute.
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.
"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?
It could be interesting to set up a mix-net rather than a single server, though.
Sure, there is no reason to have just one server.
If you simply want to experiment with blinding, for every 1.05 bitcoins you send me plus a blinded address, I would be more than happy to sign the blinded message, and for every fresh Bitcoin address you give me with my signature on it, I will send 1 coin to that address 
I would need to trust you to not run away with my money. Also I cannot estimate mixing quality you provide.