In order to answer why not a database, you have to first look at why RC uses a database.
The database is there to store users, contacts, and most importantly mail messages. Roundcube will log-in to the IMAP server, and if caching is enabled, download the messages to the database. In that database, all your user preferences are stored as well. Things like timezone, HTML preferences, display name, contacts, and others.
Now, take the database away. What if you don't have an LDAP server to store your contact information on? You just lost your way of storing contacts, so we lose one major feature. How about storing preferences? Yes, you can store it via a cookie, but how secure are they? And if you clear your cookies, you have to re-setup your preferences. And finally, speed. What if Roundcube always had to contact the IMAP server in order to get the messages (whether just the headers or entire message) just to view it, or view the list. If you have a huge inbox, it can take quite a while.
So can you see why we use a database? Can you see why not using a database would be a hinderence to Roundcube?
SQLite isn't really a server-less database. SQLite is just a performance enhanced database system.
SQLite is an ACID-compliant relational database management system contained in a relatively small C library.
You still need the physical storage of it, but it's how it interacts with the storage unit that is different. But you're still more than welcome to try it.