Wait up. I'm seeing a better picture now, I think.
I thought you were talking about putting these wrap terminals on the back of the black box to replace the connectors, but now I'm seeing a picture in my mind where the black box sits in a cabinet, the cables from black box to the machine come out a hole.
In my day job we've done some big equipment installs and we do break out cable into the racks before it goes into the equipment to make chopping and changing easier. Usually we have either a Krone punch-down terminal panel (not suitable for these cables) or a DIN rail containing a bunch of terminal blocks with screw-down clamps onto the cable ends which in turn are ferrule crimped to prevent stray wire strands causing shorts and to stress relieve the cables too.
This approach is more so that, if equipment has to be changed over and the new gear has different pinouts for example, it's easier to swap things around without chopping at the fixed (in the wall) wiring.
But adding that level of patchability is a shedload of work to do reliably and worse than pointless to do in a dodgy fashion, you don't have fixed wiring in your case and it's all pretty local. So I'd still (and I have on my machine) just stick with the extended cables coming directly from the Y chain back in through a hole in the cabinet and into the back of the black box.
KISS!