Set Field [Globals::Picker ; value:TextColor (Globals::Picker ; RGB ( 0 ; 0 ; 0 ) ) ]
Set Field [Globals::Picker ; value:TextColorRemove (Globals::Picker ) ]
Set Field [Globals::Picker ; value:TextFormatRemove (Globals::Picker ) ]
Will all remove the color, though TextFormatRemove strips out all formatting so that may be a bit drastic here. Any formatting specified in the Inspector for this field will remain in effect.
I thought I had tried similar things, but will give them another shot. I don't know if I have tried all three in succession. :)
Any one of them should work. Make sure that you don't get any extra  in your set field step. If you do, you haven't set up the script step correctly.
When Setting up Set Field, there are two Specify buttons that must be clicked. To get Set Field [Table::Field ; Expression], add set field to your script and click the first button (specify target field). Select Table::Field from the list of fields. Do not click the specify button next to the repetition box. Click OK to close this dialog box. Now click the lower specify button (calculated result) and create the expression to the right of the semicolon (;). Do not try to type in the semicolon.
Doh! My bad. I had originally used "Set Field By Name" in an attempt to make the script more flexible. I had failed to include the table name when specifying the field name, so that step was failing, thus the text wasn't being set.
Works now. Thanks for the pointers.