What I really like about doing trashed interiors is that the results are really noticeable, but they are a lot of work. Physical work.
I do not have an extractor, but here's my take on it.
Interior cleaning with an extractor more enjoyable? Probably not much, maybe a little. Depends on your definition of enjoyable. It is shiny, makes noise, has lights and switches, sprays hot water, and makes you look more professional.
Will the process be faster and easier? I would think it would be a little faster and a little easier.
It will still require a decent amount of physical work and time.
You should still agitate and vacuum first. Correct?
Still need to apply a cleaner and agitate?
Instead trying to remove the cleaner and dirt with towels, or "wet moping" plus towels and/or a shop vac, you work the extractor over the area. Which is still work.
You may still need to blot or rub the area with towels to dry it better unless you've got the convenience of a shop and are holding the customers car.
I think an extractor would improve results on nasty interiors, and may cut down on a little time.
Do you have you own shop, mobile, both?
Do you do a lot of thrashed interiors?
One thing I can think of that may make you enjoy it more is charge more. Interior restoration is a lot of work. Make the price worth what you are doing.
What type of cleaners are you using?
Since I've started using citrus solvent cleaner/degreaser type products I enjoy the process a little more. They work well, quick, little agitation needed, no foam, little left behind, dries quickly, and the smell is not bad.