from your description and what i can see in the videos, it appears you have setup all the hardware setup correctly.
1) ExPORT MODE: PRO DJ LINK is a communication protocol that is built into all PRO DJ LINK capable devices and establishes a network via your unmanaged router/switch and ethernet cables. when RB running on your PC and is in ExPORT mode (upper left hand drop down of RB screen) and connected to PRO DJ LINK network (which you appear to be) you can use your RB sort of like a giant usb/pen drive - and that's it.
2) PERFORMANCE MODE: when you set the CDJ's Source as CONTROL MODE USB A (or B) this means you are in PERFORMANCE MODE and need to change RB from EPORT to PERFORMANCE. pre-CDJ-3K every device had to be connected via USB in PERFORMANCE mode. one barely talked about new feature of the CDJ-3000s and v10/A9 is the ability to more easily use one or more players in PERFORMANCE mode as long as there's a single usb cable from DJM to computer - ethernet cables handle the rest, so no need for multiple USBs. but CDJs in USB CONTROLLER mode are not actually using the PRO DJ LINK network, just the ethernet connection that is facilitated by the DJM. the reason your A9 is showing up as a generic device is (I assume) because to use any device in PERFORMANCE mode you need RB to be in PERFORMANCE mode - in which you wouldn't be on the ExPORT mode screen (since you would be in PERFORMANCE mode screen), even if LINK mode (Export mode) is activated.
3) Also, the PRO LINK BRIDGE software is meant officially to link PRO LINK and 3rd party visual/lightning solutions and ONLY reflects PRO DJ LINK-enabled and connected devices. it's useful as a diagnostic tool in a crunch to see how PRO LINK is working, but it won't tell you anything useful about devices in PERFORMANCE/USB controller mode.
CONCLUSION: all your super expensive PIONEER DJ hardware is probably working - and doing new things you didn't even knew existed - and any confusion your having is most likely Pioneer DJ's historically abysmal ability to educate (and usefully name) users on what their hardware is capable of and under what circumstances, etc. (old news, but still wtf).
here's a super simple cheet sheet: in EPORT MODE click LINK in RB and unplug the USB B cable connected from the mixer to the PC (and use CDJ's to browse your RB). For PERFORMANCE MODE put RB in PERFORMANCE and plug in the USB B cable from mier to the PC and use CDJs and PC screen to play.
best of luck