I'm wondering it it's mostly a different needle plate and the attachments that make the difference. I've found the needle plate that has the ability to change the center part where the plate adaptor goes. And then there is the part that I assume goes in the bobbin area.
I did find one site that said the one adaptor fits 16030,17011,17012,17030,17031,18022,18023,18030,18 031
The two listed here so far are 15570 and 17550
Here is one that will chain stitch - if you zoom in on the first picture, you can clearly see the long narrow plate that fits into the needle plate - so at least I know what to look for.
http://www.ebay.com/itm/200980927365...84.m1423.l2649
Looks like Kenmore made a whole lot of slightly different models -