gcrank1, you beat me to the question about what is "FTE". Extraction and ejection are two very different things accomplished by different parts. I'm going way back in time here, to point out that the function of the extractor is to pull UNFIRED rounds from the chamber. It's a blow-back action. The same pressure which expels the bullet from the barrel blows the empty case out of the chamber forcing the bolt backward. The case then hits the ejector which kicks the spent case out of the ejection port. The opinion was expressed here many years ago that a 10/22 will run just fine without an extractor. I tried it and it worked.........when I fired the gun.
Could the issue be bolt travel? If something is retarding it so it's not moving far enough or fast enough, it could cause stovepipes or what appears to be complete failures to extract. Or maybe I'm just blowing smoke.

See below.
I have a 10/22M which experienced 50%+ failures to eject with a well-known brand after market barrel. It was suggested that I try a Clark Custom barrel and, with no other changes, the gun functioned perfectly. To me, that points to the chamber.
The 10/22 can be a quirky gun and simple things can cause issues. Like the retaining clip on the hammer spring being installed upside down. Myth or not?
https://www.youtube.com/watch?v=m7z_kZKO6Xs
Hope you get it figured out.