The standard build of SQLite uses an OS, but there is a compile-time option to omit the OS dependency. It then falls to the developer to implement about a dozen methods on an object that will read/write from whatever storage system is used by the device. People do this. We know it works. We once had a customer use SQLite as the filesystem on their tiny little machine.
Likewise, the use of malloc() is enabled by default but can be disabled at compile-time. Without malloc(), your application has to provide SQLite a chunk of memory to use at startup. But that is all the memory that SQLite will ever use, guaranteed. Internally, SQLite subdivides and allocates the big chunk of memory, but we have mathematical proof that this can be done without ever encountering a memory allocation error. (Details are too long for this reply, but are covered in the SQLite documentation.)
Finally, we do have proof that SQLite can continue after a power cycle - assuming certain semantics provided by the storage layer. Hence, the proof depends on your underlying hardware and those methods you write to access the hardware for (1) above. But assuming those all behave as advertised, SQLite is proof against data loss following an unexpected power cut. We have demonstrated this by both code analysis, and experimentally.
So probably you were correct to write your own database in this case. My point is that SQLite did not miss your requirements by quite as big a margin as you suppose. If you had had a bigger hardware budget (SQLite needs about 0.5MB of code space and a similar amount of RAM, though the more RAM you feed it the faster it runs) then you might have been able to save yourself about two years of development effort.