Yes, the salt cell breaks apart the sodium and chlorine. However, you still need add chlorine shock occasionally, depending on weather conditions. I have to shock mine at least once a month during the middle of summer and twice a month in spring and fall. Rain can create algae pretty quickly.
The sodium builds up in the water and you have to add acid to keep the alkalinity down. This creates... salt, and the process starts over.
I have had a hard time finding chlorine shock for the last month. Even the pool supply companies are out.
So, in chlorine generator, the sodium from the NaCl is released into the water & reacts with the water, forming NaOH & hydrogen. Then you have to add acid to counteract the sodium hydroxide.
Hmm, if you add HCl as the acid, you get water + salt (NaCl). You would never need to add salt with this scheme, just HCl (hydrochloric acid).