I don't think private lessons will work out. There's probably some short term revenue but I doubt it will last. I remember a few WC players attempted something similar and they charged like $50/hour for lessons. I also don't think it will help in introducing people to the scene either.
I like the idea of tutorials but it shouldn't be paid. I believe top players should be creating content on Youtube (instead of streaming on Twitch). Streaming generates revenue in only one time period whereas youtube videos are persistent. Top players could create tutorials, provide match analysis, help out viewers or have some long sets with commentary. That's just one part of the formula. The players themselves need to be able to retain viewers by being entertaining, wellspoken, charismatic, etc. Crosscounter did an amazing job and were able to not only generate revenue but introduce people to the scene. Content creators could use CC as a guideline of what they should and shouldn't do and build on to it.
As a person who has done paid tutorials for SF4 before, I'll chime in.
I remember that I toyed with the idea and spoke with my stream viewers about it. I was apprehensive because I didn't like the idea that I can sell something which has no real quality guarantee (and I'm a bit of a perfectionist, plus I've worked a lot in customer service). However my stream monsters insisted that I give it a try and so I did.
I charged $30 for 1 hour, or I allowed two separate hours to be scheduled for $50. You can do both hours at once if you like. I settled on this price because my viewers said I should treat it like music lessons, which go for about the same price or often higher.
My goal in a lesson is to ascertain your skill level and give personalized advice to help you meet your goals. In the instance of multiple lessons being paid for, I would check your progress by asking questions about previously covered topics as well as playing certain styles to see how you improved in the interim.
I would agree that this is not a way to bring in new players. However I think that the personalized lesson approach can be a great fast-track of improvement for intermediate players who are trying to reach upper-level play. Having a competent instructor is also key, and I wonder if other players who have given lessons gave it this same amount of forethought. I also thoroughly enjoy teaching in general and almost pursued a career in it, though I suppose I still haven't completely ruled it out.
I also agree that top players should seek to create and distribute content (I am actively working on KOF stuff), but I recognize that this requires a certain personality type and knowledge level to make effective. Sirlin wrote about it in Playing to Win, when near the end of the book he commented on how to approach your tournament success once you've attained it. As the FGC grows content of this type, especially tutorials, will be easier to find, but currently there is a distinct lack of effort in this area especially in games with smaller communities.
As with almost everything else in the FGC, it comes down to the fact that we are a niche within a niche.
