The steam deck doesn't auto-mount external hard drives (you can confirm this by going into desktop mode and seeing the prompt to mount and browse the drive contents). Normally, SteamOS only initializes the internal SSD and the SD card currently inserted (if it's in the file format that SteamOS expects, they provide a tool in settings).
You could purchase an SD card, place it in the slot, change over to Desktop mode, mount your external drive, and then copy your games over using steam (it's even a feature outside of SteamOS - offline transfer between storage). That would probably be the best option.
The way Valve set up the Deck is that the only storage initialized is the internal SSD and the SD card in big picture mode (home menu) which makes sense if you're playing on the go - you're not going to hold an external HDD in your offhand on the plane lol.
I believe using the dock doesn't have any impact on how the Deck handles drives either, the official FAQ mentions it's more akin to "plugging an external USB hub", so that won't solve the automount issue.