The good news is the router has USB streaming capabilities. The bad news: it doesn't work.
I do understand the External HD should appear in the Network area. I can see My computer, Media Devices, Network Infrastructure ( Cisco) and Printers ( wireless printer). I used to be able to see my HD with previous router a few seconds after I would plug it in.
I'm not sure how you determined that the USB port on the Cisco DPC3825 has any useful function at all. I don't want to be the bearer of bad news (please don't shoot the messenger ) but nowhere in any Google search, or in any reading of the official Cisco documentation have I been able to find that the USB port on the DPC3825 can perform any useful function of any kind, at least not one that Cisco will admit to. The documentation says that the front panel USB led will flash under certain conditions of data transfer, but it fails to specify what can be connected to the USB port to make it transfer data!
I understand your desire to connect an HD to the USB port. I can do that with my Ericsson W35 RocketHub and find it useful. On the W35 the USB-connected HD shows up as a Windows networked drive.
In the Ericsson W35 Management Interface Web pages there is one page which details the setup and configuration of shared Windows devices: hard drives and printers. The Ericsson W35 uses a variant of Linux as its O/S and has the SAMBA package included in it for this function. In the Ericsson "Sharing" page, you can specify the hostname to be used, and the workgroup of which the device is to be a member. Further, you can specify the share name for the networked HD/USB stick, or whatever, and also you can specify the name of a connected shared Windows printer.
None of the above is to be seen on any of the Cisco documentation, or on the device management web interface, as far as I can tell.
So, unless I am very wrong (and it wouldn't be the first time!), and have missed something, I doubt that the DPC3825 actually supports anything useful connected to its USB port.
My apologies if I have misread the situation. And it is always possible that Cisco will come out at some time with firmware updates to augment the DPC3825 functionality in this area.
I do understand though that the DPC3825 is not actually a Cisco-made device. It is actually produced by Scientific Atlanta. So, who knows what the future holds there.