I think the real benefit of using the extractor over hand shampooing and wet vac is the amount of suction you get from the extractor. A lot of vacuums you get at home depot, even the big hp 16 gallon ones, have like 60" of water lift, while most extractors have well over 100".
Hand shampooing for me goes pretty quick though, because the volume of water and cleaning product used can be laid down quicker and heavier than an extractor. Sometimes its not a great thing bc of oversaturarion, wicking etc.
I think chemicals make a bigger difference than the machine. If you pretreat the carpet with a good pre spray before you lay down your shampoo, and lay down a fiber rinse product to neutralize the alkalinity of the cleaner, then extract, the carpet and upholstery will come out clean and soft, regardless of the machine