I think SQLite is fantastic but it does start to fall apart at the edges sometimes.
What is more interesting to me is the fact that everyone seems to think of Postgres as the obvious alternative to SQLite. It is certainly an alternative. For me, the most opposite thing of SQLite is something like Oracle or MSSQL.
The complexity being relatively constant is the part I care about most here. Running a paid, COTS database engine on a blessed OS tends to be a little bit easier than an OSS solution that can run on toasters and drones. Especially, if you are using replication, high availability, etc.
The business liability coverage seems to track proportionally with how much money you spend on the solution. SQLite offers zero guarantees accordingly. You don't have a support contract or an account manager you can get upset with. Depending on the nature of the business this could be preferable or adverse. It really depends.
For serious regulated business with oppressive audit cycles, SQLite trends toward liability more than asset if it's being used as a system of record. That it merely works and performs well is often not sufficient for acceptance. I'm not saying that Postgres isn't capable of passing an intense audit, but I am saying that it might be easier to pass it if you used MSSQL. The cost of having your staff tied up with compliance should be considered when making technology choices in relevant businesses.