This isn't really a new approach to scaling--even if you're using a server DBMS like postgres/mysql/etc. you still have to figure out scaling based on your data. If that data is easily isolated into customer/user units then it's straightforward to shard them across more instances, whether they're more postgres boxes or SQLite processes. If the data isn't easily sharded... well you're no worse off whether you're using SQLite or any other database.

The main point here is that instead of reaching for a costly to maintain, to configure, to secure, etc. DBMS first, many folks would likely be better served by the vastly simplified management of a single SQLite process.