I finally found a solution.
Even though EVERY tutorial on the Internet says that you need to connect the contrast pot to 5v, for some reason, the pin3 is powered internally for this specific LCD, so all I had to do, was to not connect the pot to 5v.
See attached picture.
P.S. thank you david_prentice, I saw your comment after I have found the solution.
You don't actually need to connect data pins to see if contrast works. As for for A and K pins, they control the brightness, not the contrast. But thank you anyway
