You can make up as many fictional backstories as you want, but gmail was launched because signed-in users enjoy vastly better search quality and prior to gmail there was no reason to associate your identity with Google.
Is that view incorrect?
My overarching view on Google's evolution is that their real product is dirt-cheap computing facilities, and their products are often just a way to exploit that capacity. Part of Gmail's genesis was an abundance of underutilized hard drives. YouTube was acquired because they already had the networking in place to operate it. Inbox Smart Reply was a way to use a novel and cheap ML inference device to launch a feature that nobody else could afford. In all these cases the infrastructure existed first and the products emerged to fill them.
Email search wasn't the draw initially, and wasn't even all that much better; there was serious controversy over search, years later, as it was tied to ads that reflected the content of private communications.
Hotmail, Yahoo et al had reasonably good search for the time, but stiflingly little free space. Google offered far more free space with IMAP and POP3 access.