Hexleg, I love your blog, I hope mine looks half as good when I finally get it up and running!
I did call the Appliance Service Center - they came out, looked at it, said it would be 5k to fix if they could find the parts without telling me WHAT was wrong with it, and charged me a rather exorbitant service call. I had high hopes for them based on that ad.
I found another local service place that would work on my stove, and for an $80 service fee, they a)cleaned it out, b) showed me how it worked, hooray (how to adjust the amount of gas going to each burner, how to clean it out, basic function things), and actually TOLD me what was wrong with it. I need new hinges, a new thermostat, and some new knobs & such. He's been looking for parts for me for months, but isn't having any luck, so I've turned to the internet.
Don't mind the mess in this photo as we are under heavy construction in the house and nothing is even close to finished! Here is an image of my baby when we first got her in, pre-cleanup. You can see the broiler unit on the left, the oven on the right with the thermostat that needs replacing (damn you, Wilcolator type S!), and in the middle where the "griddlelator" would have gone is additional cabinetry. Apparently, you used to be able to get these things configured any way you would like and the folks who had this built didn't want the griddleator.

Clearly, she needs some work, but the oven repair guy that actually talked to me said she would easily run another 70 years and that I really had a gem:

Your stove is clearly is a real beauty! I hope to get mine into that kind of condition soon.
"Baseball is like church. Many attend. Few understand." Leo Durocher