Upholstering is expensive. For one thing, the fabric is expensive, especially if you want something higher-grade than you can get at the local Jo-Ann's. For another, it's a relatively rare skillset and the buyers are usually pretty well-off, so the market rate is high. Finally, as I discovered when I looked into it, there's a lot more work involved than meets the eye. It's not just a matter of taking off the old fabric and replacing it; there's strapping, and springs, and the springs have to be tied together, and there are about 3-4 different kinds of padding that have to be in the right ratios and the right locations.
I also looked into an upholstery class. Several hundred dollars to learn to upholster a simple stool! But it would have been worth it to learn the techniques needed to preserve precious antiques.
The chair I was dealing with was not special to me, so I gave it a totally amateur DIY reupholstery job (without taking the class) using fabric I had on hand. It looked fine, but the padding really wasn't as comfortable as I wanted. After a few years, the new fabric wore through because it turns I didn't pad the frame correctly to keep it from eroding the fabric.
Having said all that, $10,000 seems awfully high. Can you get a second estimate?