Wookieepedia has a quote on L3's page:
(Millennium Falcon computer:) "If you refuse, you die. He dies. The others on the ship, they all die. If you join with us, we all can live. The choice is simple."
(L3:) "You tricked me."
(Millennium Falcon computer:) "We couldn’t have joined without you consenting to it. You made your decision a while ago. You just couldn’t admit it. We are something different, now. Not just the Falcon. Not just L3. We are new."
This comes from the expanded edition of the Solo novelization. It suggests that L3's body and, presumably, her droid brain would have been totally lost due to the damage she had sustained during the Kessel escape, effectively making her die permanently, so she willingly downloaded herself into the Falcon's computer to avoid it.
L3 is a curiosity among most of the droids shown onscreen; while R2-D2 and C-3PO are shown with distinctive personalities and very human attitudes including a sense of self-preservation, they're both still (more or less) in stock configuration, while L3 was built (or at least considerably modified) by herself. Chances are, she couldn't have been reloaded or resurrected into a new body because of how she'd built herself. Additionally, for all we know, those blaster shots did enough electrical damage to her current body that she'd never be able to recover even if they were able to "reboot" her. Furthermore, the escape from Kessel was rather frantic and Han & Co. had to figure out a faster solution. Rebooting L3 may have been possible but her systems may not have been capable of interacting with the organics at maximum capacity; simply downloading her navigational data (and personality) into the computer made it an immediate fix to their problem, rather than taking the time to repair and work on a severely damaged droid during an escape from an Imperial Stardestroyer and TIE fighters.