And If I'm legally allowed to live in Poland, then I feel that I'm legally allowed to be in Schengen zone.
No, doesn't work like that. The stamp is only recognised as a means of staying legally in Poland, not in Schengen. You need to fly directly from Poland to a destination outside of the Schengen zone.
I essentially don't know if the Karta Pobytu application process in Poland has been propagated throughout Schengen zones
No, it hasn't. You're only legally allowed to remain in Poland. In other Schengen countries, you've overstayed.