The answer is that there are multiple layers to any given musician's music, and while some are easy to duplicate, others are impossible. Also, some layers can be easily duplicated by a listener, and while other layers might not be, it still could be in the artist's interest to provide that information.
As an example, after just a single listen (and another to check), I can come up with this transcription of "le riff":
This is nothing more than the notes and rhythms, which in this case are fairly easy for a trained listener to transcribe. (I don't know harmonica tablature, but it would be isomorphic to this notation).
The notes and rhythms are usually seen as the top level. They can be easily "figured out" and duplicated by others, but this is only one piece of the puzzle. As the instructor mentioned in the video you referenced, there are subtleties in playing the riff that are below the surface: producing the same pitch in a different way for ease of technique, or a different tone, and exact length of bends that aren't as obvious because they go by very fast.
Another level down might get us to the original artist's hardware and microphone/recording setup, i.e. the how of replicating the artist's unique tone. Eventually you get to the artist themselves -- their unique physiology that can never be replicated exactly, but still has an effect on the overall music that they are creating.
Whether an artist decided to share any of this information that is known to them is highly dependent on their personal feelings about copyright law, how they make money to begin with, and art in general:
- An artist who owns the copyright on a song might license it to a publisher so they both make a ton of money on royalties for sheet music, karaoke, etc.
- A classical composer (dead or living), makes an income off of sales of the sheet music, and royalties from the performance of that music by other people.
- A guitarist might reveal some of their individual playing technique and practice tricks to make an instructional video to be sold.
- Any instrumentalist might partner with an instrument manufacturer to create a "signature line" of instruments that they use exclusively, in order to benefit both parties.
At the same time:
- Early jazz masters (at the dawn of the recording industry) didn't want to be recorded for fear of other musicians learning to play their music and taking away their business (of live concerts). The way you learned to play jazz was by going to live concerts and then trying to imitate the masters in your practice room.
So, to recap and answer your individual questions:
People might have get their hands on the music of the dead musicians like Mozart, but what about people like Slash, Joe Satriani etc. Do they share the chords?
Mozart's not just dead, his music is also in the public domain! He and present-day classical composers depend on their music being played by other musicians, so they copyright a published product containing the notes and chords, and then sell that. Slash and Joe Satriani are recording artists who make the majority of their income from record sales and live concerts. Most of what they play is very easy (relative to all of music, that is) to figure out on the surface levels, but if you got on stage and played all of Slash's notes and rhythms, you wouldn't sound like Slash. Joe Satriani is largely the same way, although I believe a larger percentage of his music is stuff that can't be easily figured out just from the recorded sound alone -- you might need to see him perform live to get a clue!
You'll probably find that people are more likely to publish their notation when the music they are playing is idiomatic to styles that people are already familiar with. When you are composing AND performing new music that sounds like nothing anyone has ever heard before, then understandably you might not want to reveal all of your secrets!
And lastly, I don't think you need to separate out vocalists from the thought process here. The only thing they are really sharing with the band is an arrangement -- something they may or may not have written themselves, or may only be a structure and key. It is impossible for a vocalist to share their actual voice, which is usually what distinguishes them from all of the other singers out there.