Is it possible to fix some of these limitations by building DBMSes on top of SQLite, which might fix the sloppiness around types and foreign keys?

Using the API with discipline goes a long way.

Always send "pragma foreign_keys=on" first thing after opening the db.

Some of the types sloppiness can be worked around by declaring tables to be STRICT. You can also add CHECK constraints that a column value is consistent with the underlying representation of the type -- for instance, if you're storing ip addresses in a column of type BLOB, you can add a CHECK that the blob is either 4 or 16 bytes.