There are a number of potential problems but to sort through them may be tedious.
First, the new alternator may be defective or undersized, and unable to put out the necessary voltage to trigger the dash lamp response. In this case the battery is still providing juice and you may wear down your battery.
Next, the connections may be less than ideal leading to Voltage drops from the alternator, which will also give you the response you see. Was anything covered in a preservative on the new wiring harness? If so it may need to be removed to make a good contact.
I can't remember, but how is your battery? If starting is really stressing out your battery, it may be drawing so much current to recharge the alternator just can't keep up. You did not specify where you got the rebuilt alternator. I would suspect it may be under rated for this service.
What is your idle speed? It may be that the car has a really low idle speed and the alternator just does not have the speed to make the power at idle when it is also recharging the battery.
By the way, once you get to 13.8 volts by revving the engine, you say it stays there at idle. Do you mean fast idle or normal idle?
Good luck, Jim
1986 Euro 190E 2.3-16 (291,000 miles),
1998 E300D TurboDiesel, 231,000 miles -purchased with 45,000,
1988 300E 5-speed 252,000 miles,
1983 240D 4-speed, purchased w/136,000, now with 222,000 miles.
2009 ML320CDI Bluetec, 89,000 miles
1971 220D (250,000 miles plus, sold to father-in-law),
1975 240D (245,000 miles - died of body rot),
1991 350SD (176,560 miles, weakest Benz I have owned),
1999 C230 Sport (45,400 miles),
1982 240D (321,000 miles, put to sleep)