Made a ramp out of the backyard concrete tiles, got under the car and found out what it was.
One of the two transmission shift linkage bushings is completely gone. The one at the shifter stick looks fine, the one at the transmission gear box level has vanished.
Here are some pics for anybody who runs into this issue.
Assuming this is all Mercedes SLK320 type, it looks like this is what I need (available from partsgeek.com):
Vaico Automatic Transmission Shift Linkage Bushing
Part Number: 6500-01148972
Typical $2 part that is going to take me a day to replace because there is so little access...


Green bushing on the shifter stick side still there.

Bushing at other end of linkage gone.