"a cloud storage drive requires very good internet connections and fast response times."
No, that isn't the issue. The QB desktop program does not load the data file into memory. Word and Excel put the working file into memory, so you are not Working on the live file.
QB requires a Live and Managed connection to the database at all times. Think of this as a phone call that you cannot Ever set it down or put it on hold. There are associated utility files that are Live Updated while you work. That's why there is a Save on every Transaction; not a Save on the entire file, at the end.
"Some programs don't like anything but local disk access (and it appears that Quickbooks may be one of those)."
It's not an issue of Speed; the data file must be Managed. The only cloud service provider doing this so far is QBox.
"Is there any reason this will not work?"
You will have a couple of issues. Mostly, that device is really slow, compared to the local harddrive, for the relational database and the associated utility files that are Live Updated while you work. This is not like Word or Excel, in that the file is not Loaded into local memory while you work.
"I will use a cloud service to backup the hard drive. "
That also isn't good enough. What you would do is use the QB function to make a backup file (.qbb) and you can make that to the external device. Then, the Cloud will mirror the .qbb. The .qbb is your safekeeping and disaster recovery file.