I have a Bighorn TL3 that I’ve had for about 3 years now and I’ve always had some sort of extraction issue with it. I was sent a new extractor with spring and then a new bolt head neither of which seem to work. I’ve had two different gunsmiths one being patriot valley and the other a local gunsmith make me several barrels for it but I still have the same issue with having stuck cases. Has anyone else had this problem? My loads are pretty mild for 6 BR and 6 Creedmoor but I’m still having this issue.
The only time I've ever had a case slip off the extractor in my TL3 was when I had a very stuck case (fired brass shot in someone elses rifle). It was stuck to the point of having to shove the bolt handle upwards with full effort, and it slipped off the case head when it actually deformed the brass in the case rim, essentially tearing a ramp into the brass.
I'm going to start with the assumption that you aren't having any primary extraction issues, ie. the bolt is not hard to lift when cycling, it doesn't "click" open when the case pops out of the chamber, etc. If there is any excessive effort to get the case to pop out of the chamber, then you can look into the issues with headspace, base diameter sizing, making sure the load is lower pressure, etc. All those fixes would reduce the amount of bolt lift effort and extraction effort, reducing the amount of force on the extractor.
If the above isn't the issue, then I would look closely at the mechanical function of the extractor. It needs to be able to engage over the case rim as far as possible. Check the cleanliness of the extractor groove in the bolt head and detent/spring to make sure that it slides left/right freely, maybe even put a drop of oil in the mechanism. Then I would check the thickness of the case rim relative to the amount of space between the extractor and the bolt face - is the rim too thick to fit underneath, causing the extractor not to fully engage over the top of the rim? Finally, I would look at the underside edge of the extractor, making sure it's sharp and square to the bolt face and not rounded or angled in such a way that would allow it to slip over the case rim rather than biting into it.
For reference, on my TL3 the extractor sits about 0.050" inset from the side of the bolt head when there is no case underneath it (measure using the butt end of a caliper, from the outside edge of the bolt head to the outside edge of the sliding extractor). When I clip a piece of Lapua BR brass under the extractor, it is still pretty much fully engaged, measuring about 0.045 to 0.048" inset. Let me know if these numbers don't make sense and I can take a picture.