It is certainly possible, though, to please most people. The date types in other competing SQL engines are widely used; I don't think you get off the hook by just listing a couple of different requirements that people have and saying that it's hard and throwing up your hands. DATETIMEOFFSET is not possible to reasonably implement in SQLite using any of the built-in types because you need special equality semantics that understand how to convert time zone offsets. You can't just pack it into a string or a number and hope for the best; comparisons between values don't work out. There's no way to pack it into a string format in such a way that a regular string comparison will provide chronological ordering and correct equality semantics.
I have built an entire T-SQL layer on top of SQLite and I had to omit DATETIMEOFFSET support entirely. There's no good way to do it with the tools that SQLite provides. Please let me know if I've missed a way; I would like to add this to my extended SQLite language. DATETIMEOFFSET is the gold standard in the SQL Server world and it is a gaping hole in my compatibility layer.
I have also built a virtual table module that links remote SQL Server tables into SQLite. I had to discard the time zone information for DATETIMEOFFSETs in the remote table and show it in SQLite as a UTC time only in string format. This is strictly worse than what we have in SQL Server, and the gap cannot be crossed with clever application-side code.