I think I may have found the issue with the power on circuitry on the Betz board. Your notes about how it behaved in unexpected ways after you changed the capacitor I don't think are related to my case because I didn't change the circuits on the board.

The BlueGiga dev board has a much more complicated circuit related to power on. Located inside this zip file is the full schematic for the BlueGiga dev board and it has a very specific note on this:

https://www.silabs.com/documents/login/reference-designs/DKWT32i-v2.2.zip

It says "Minimum voltage hold circuit to prevent reset causing turnign [sic] off the board. This circuit keeps the enable pin active until FW has booted after the reset."

Whereas the Betz board simply has a switch to keep the Enable pin always active. I wonder if that's the root of the problem, that the more complicated circuit is needed?