My question was, do use the existing spindle ground cable to ground the machine, or add a second, dedicated cable from frame to star ground (meaning, if we consider the spindle ground, there would be two paths to the star ground point from the frame.

An example may be a gantry machine where the gantry has the spindle motor mounted on it and has its normal ground conductor in its supply cable as you have mentioned.
This spindle may be on a slide for Z axis operation.
In this case, It also pays to ensure the gantry itself is properly earth bonded to the rest of the stationary part of the machine via a separate GND cable.

 

From your star, ground back to source receptacle only.

 

Here are two examples where I used a ground strap for 2 different reasons:
1: RF matching network to chamber. RF comes in on an RG-8 cable. Chamber gets bonded to matching network with a braid. I found out later that you can buy braid with eyes on them.

2: TV antenna: Ground strap around rotor. Static and lightning. I don't have a ground rod and I should. There is a tree that wuld stand to get hit first, so it could be a "lightning rod". The mast is attached to Aluminum siding near foil backed rock like "sheetrock". The antenna is bonded to a grounded outlet about 30' from the antenna. About 15' is outside.
It's really hard to come up with "how". You have a chimney and a driveway along the entire eve of the house where the antenna is. Then the driveway extends maybe 15' along a shed roof. Footing is probably 4' down in the ground. You don;t want to crack it and you don;t know how far it extends. I need the static protection and the tree will likely get hit first. It's probably 30-40' feet above the antenna. The antenna is guyed with insulated guy rope called Phillystran. If I have the opportunity to change out the antenna, I'd use a fiberglass mast on the antenna section.

 

The OP's concern seems to be in the area of Industrial control, CNC applications in particular..
There is also NFPA79, an older version avaialable out there in PDF