To answer my own question, I read this quote from Steve Brotherton in his Evaluating Electronic Engine Controls
article in the DIY section of this site.
The OVP also totally fails or burns its fuse, killing all EHA current. This condition doesn’t affect a warm motor, as it should already be running closed loop at zero milliamps. But it causes most of the hard cold-starting problems.
Which certainly seems to indicate that a faulty EHA will cause hard cold starting because no enrichment will be available.
If I have time before my new EHA arrives, I may swap the EHA out of my 300SEL and see if that fixes the problem. It seems that most of the gas engines in the late '80s used the same EHA. The part number of the one in my 300SEL is exactly the same as the one in my 560SL and I've noticed the same is true with the 260E and 190E 2.3.