Hi Ben...
I agree that carpet stain removal is troublesome. Often as troublesome as fabric upholstery stain removal.
It starts, IMO, with identifying the type of stain that's present. There are inorganic, organic...and I put ink stains in a seperate niche. I also put stains into one of seven categories:
1.) Grease/tar/adhesives...(inorganic)
2.) Pet stain/odor/neutralizer (uric acid)
3.) Rust stains.
4.) Coffee/tea...(tannins)
5.) Blood/Ketchup/Mustard.....(protein/organic)
6.) Red dye/Red juices.
7.) Inks.
There are companies that sell stain-specific removers....Usually a little more expensive...But by far the best route to take, IMHO.
If cleaners such as Folex, APC's, P21S TAW, etc. won't budge the stains, it's good to have the stain-specific removers handy.
-Steamers/extractors are very good tools to also have in your arsenal someday.
-Sometimes janitorial/carpet-cleaning brick&mortars hold classes in regards to this matter.
Hope this helps some.
Bob