> then what's the reason people aren't using it for websites?
I'd guess the reason to be that people keep hearing things like "don't use SQLite for websites" and thus don't even try.
> Why do you say it works for "small" websites but presumably not large ones?
Not the GP, but the main reason I wouldn't use SQLite for a large website is that SQLite itself doesn't offer much re: failover/replication (i.e. multiple servers, one database), and I haven't used RQLite enough (or at all; I should fix that) to be comfortable with it in production. Because of that, I'm more likely to reach for / recommend PostgreSQL instead.
That being said, if your website has crazy "web scale" FAANGesque needs and you're at the point where you need to write your own replicated datastore, using SQLite as a base and building your own replication layer on top of it (or using RQLite and maybe adjusting it for your needs) seems like a reasonable way to go.