Sounds to me like you did what most people do, You did what the dealer did when they installed the aux brakes on my towed, And .............
The manual says: The brake break-away safety cable should be attached firmly to the frame of the towing vehicle and NOT to any part of the hitch gear.
Alas.. You did what most of us would do.. Myself included.
Why is that wrong? I had two brackets break on my motor home, Dropped the cross member off the frame rails, this cross member is what holds the hitch receiver and, of course, the spot where you hook the safety cables.
Thankfully,, Nothing bad happend.. (Well I lost a bicycle tire and rim but that's minor, 80 bucks minor).
Had the break away cable been hooked to the motor home FRAME instead of that cross member... The car might have stopped.. Alas... That might not have been as good an outcome in my case.. But the point is the hitch receiver in my case, which is where those connection points are, came off along with the rest of the hitch.. Oh well. Better luck next towed (Which is now parked in front of the motor home awaiting the final mods for towing).
Home is where I park it!