SQLite would be nowhere near the current design in term of performance. Typical shaarli usage involve little update (a dozen a day maximum) an a lot of read (10k+ pages viewed per day on sebsauvage's instance (shaarli's creator, an influent french blogger). Thé file system's cache works really well in this kind of scenario, way better than SQLite.
For the stability guarantees, shaarli exists since 2011 and is used in production by many (several hundred) people without trouble.