I know that the salt bridge produces ions to balance out the positive and negative ions in each solution, but why won't it work without a salt bride? Would it even work at the start since there's few electrons gained and lost and few ions produced?
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.