SQLite has some advantages over MySQL, such as:
1. Lower memory footprint
2. No need to have a dedicated server process running.
3. Easier deployment: no need to install additional software besides PHP and Apache.
4. Great for local development (easier setup).
5. Easier to migrate content from the local machine / staging server to the production server and vice-versa (you only have to FTP one file, no need for data migration tools and SSH tunnels).
6. Easier to backup/restore.
7. Easier to configure and maintain (less knobs and dials to finetune).
8. More flexible schema (columns are not statically bound to a specific type) and simpler data types.
9. Has good read concurrency (enough for small/medium websites).
And SQLite does support triggers, foreign keys, multi-column and multi-datatype (per column) indexes, constraints, cascade updates/deletions, transactions, and other features (much more than people usually assume).
It also has decent performance, if the database is correctly indexed and if the SQL queries follow some optimization rules.
1. It's not good for write-heavy applications, as tables are locked when writing.
(note 1: most websites are usually read-heavy, and write to the database much less frequently).
(note 2: there is a new operational mode for SQLite that allows for simultaneous read and write but I'm not sure if it can be enabled in PHP).
2. Join performance is not as good as with other, more powerful, databases.
As for the level of demand for SQLite support, let's see what the community has to say ;-)
I just wanted to bring this possibility (of SQLite as an alternative) to your awareness, so that if the community starts asking for SQLite, you will be able to reply "Perhaps" instead of "Not possible".