Why would electrode reaction in galvanic cell stop when there's no neutrality in each cell?
It would work vanishingly briefly. The separation of charge that occurs when you connect two separated half cells without a salt bridge produces an electrostatic potential difference that opposes the electrochemical potential difference. The cell reaction would proceed until the two voltages balance, and it doesn't take much charge separation to hit the 1–2 volt electrochemical potential of typical cells, so this process is very fast.