I've not owned a mac in over a decade so don't remember if they notify the user of when a USB device is connected. I vaguely remember a device list somewhere that could be viewed, but I might be just imagining things. I do know when any USB device is connected to a Windows machine it'll let me know.
If you are trying to use it with a DAW, it should have a list of MIDI devices for you to connect to, so you should be able to see it in it's list if the BSP is actually connecting to the computer.
That's really annoying the seller didn't supply you with the Y-cable as it's pretty necessary when you are running the BSP while it's connected to a computer as it does tend to introduce noise. I've noticed on my old laptop that if I used a particular port to connect the BSP to modify setting in MCC it wouldn't see the device even though it was getting enough power to light-up. I solved it by using the Y-cable. I figured the port was not supplying the required 500mA to allow the BSP to operate correctly. That said I can't see you having had that same issue on four different computers.
Did the seller supply the MIDI adaptors? As it is powering-up it would be pretty safe to assume it's getting the required power to run, but just to be sure try sequencing something via the MIDI Out port.
I talk with someone at Arturia France on facebook and he suggest me to shake the usb cable until he recognized the BSP.
LOL! That sounds crazy but does sorta point at the USB socket on the BSP maybe being a bit loose and needing a jiggle for a good connection? It's always a gamble buying 2nd hand gear as you don't know what's happened to it while the sell has owned it. If you are in contact with the seller still maybe ask them if they had similar problems?
Best of luck.
