How do you get your musical career started as a pianist?
Well, it depends on your goals. If you're planning on becoming a freelance pianist, it might be best to become as well-rounded a musician as possible. That's the approach I took. I can play Broadway musicals, conduct and accompany choirs, arrange for and conduct orchestras and big bands, coach vocalists, teach classes, and perform in small groups in many musical genres from pop to jazz to rock and much more. This wide range has served me well over the years and has kept me very busy as a professional musician.
But what about it you're envisioning a "solo career," where you're reaching the general musical public as a solo artist? In this case, the jack-of-all-trades approach might work against you. (Keep in mind that there are no "rules" here. I'm speaking in general terms about what works for most musicians.)
It seems to me that one challenge many eclectic musicians have, as far as the musical public goes, is that the whole music business (and I include public image here) works best when we can be put into a stylistic box. Even the exceptions, like Joe Zawinul (whom I saw mentioned on your site), Chick Corea, and Herbie Hancock, all started in a "box." That box was Miles Davis, which the public embraced. And once they got famous for playing jazz rock with straight 8th notes at a time in history when this was "in the air," the public would then follow then anywhere they wanted to go, musically. Keith Jarrett too, after Miles and the straight 8ths of the Koln concert.
So it may be that once you establish a strong enough beachhead in one very specific area, you can then branch out into all the other styles you enjoy playing, and play so well.
Whichever direction you take, stay focused, work hard, and be persistent. Good luck!
Learn the 5 Essential Left Hand Techniques with my free ebook: Left Hand Techniques for Jazz Piano
You'll also get my weekly jazz newsletter with practice tips and inspiration